Research on Smarandache Unsolved Problems

Transcription

1

2 Smarandache 未解决问题研究 编者 : 李江华 西北大学数学系 郭艳春 咸阳师范学院数学系 译者 : 王锦瑞 杨衍婷 刘艳艳 西北大学数学系 High American Press 009

3 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning University of Microfilm International 300 N. Zeeb Road P.O. Box 346, Ann Arbor MI , USA Tel.: Customer Service Peer Reviewers: Wenpeng Zhang, Department of Mathematics, Northwest University, Xi an, Shannxi, P.R. China. Wenguang Zhai, Department of Mathematics, ShanDong Teachers University, Jinan, Shandong, P.R.China. Guodong Liu, Department of Mathematics, Huizhou University, Huizhou, Guangdong, P.R.China. Copyright 009 by High Am. Press, translators, editors, and authors for their papers Many books can be downloaded from the following Digital Library of Science: ISBN: Standard Address Number: Printed in the United States of America

4 ,.,.,..,,,,,,., Smarandache Sn, F. Smarandache., 99,, F. Smarandache 05.,,., Smarandache Smarandache.,.,., High American Press Smarandache.,,.,!,,,,, 08YZZ30,! I

5 Smarandache Smarandache F.Smarandache Smarandache Smarandache d f n F.Smarandache LCM Smarandache Pierced Smarandache Smarandache Smarandache F.Smarandache k F.Smarandache Smarandache 36. Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache II

6 k k k Smarandache Zwn Smarandache Zwn Zwk θk Smarandache Smarandache F.Smarandache LCM Smarandache Smarandache Smarandache F.Smarandache Smarandache Smarandache ax by + c = ax by + c = N Smarandache Smarandache Andrica Andrica Smarandache Christianto-Smarandache. 9 III

7 Smarandache 7.7 Smarandache Smarandache G,, Smarandache Ceva Smarandache Smarandache Smarandache Smarandache Smarandache IV

8 Smarandache Smarandache. smarandache,. Smarandache, F.Smarandache, F.Smarandache LCM. 6. smarandache.. F.Smarandache n, fn F.Smarandache, f =, n > n = p α pα pα k k n, fn = max i k {fpα i i }. Sn = min{m : m N, n m!} F.Smarandache. Sn, n = p α pα pα k k n, Sn = max i k {Spα i i }. Sn F.Smarandache, S =, S =, S3 = 3, S4 = 4, S5 = 5, S6 = 3, S7 = 7, S8 = 4, S9 = 6, S0 = 5,. Sn,, [], [3], [4] [5]., Farris Mark Mitchell Patrick [] Sn Sp α. p α + Sp α p [α + + log p α] +. Lu Yaming [3] Sn k, Sm + m + + m k = Sm + Sm + + Sm k

9 Smarandache m, m,, m k. Jozsef Sandor [4] k, m, m,, m k Sm + m + + m k > Sm + Sm + + Sm k. m, m,, m k Sm + m + + m k < Sm + Sm + + Sm k. [5] Sn,. : P n n, x >, : Sn P n = ζ 3 3 x 3 ln x x 3 + O ln, - x ζs Riemann zeta-., n = p α pα pα k n, k SLn = max{p α, pα,, pα k k }. F.Smarandache, F.Smarandache LCM.,, [6], [7] [8]. F.Smarandache Sn, [9], Sn :S =, n > n = p α pα pα k k n, : Sn = max{α p, α p, α 3 p 3,, α k p k }. F.Smarandache.,, [6] Sn Sn. [9] : k. x >,

10 c =. n= Smarandache x >, = e c x ln x +O xln ln x ln x, n N Sn x lnn + nn +.,. x x >, p x, α = αp αp p x < αp + p αp = [ ] x. p Sn n = p α pα pα k k n, Sn x d n Sd x., m n m, n = Sm x, Sn x, Sn [ Smn x. αp = x p ], m = p x p αp, Sn x n m. N, n N Sn x = d m = dm = p x + [ ] x p = e p x k = [ln x], [0] [] πx = = x x ln x + O ln x p x ln + [ ] x p. - ln p x + [ ] x p = ln x ln <p x x + [ ] x p + p x ln x ln + [ ] x p 3

11 c = n= Smarandache = = ln x ln <p x x k n= x n+ <p x n + [ ] x p ln + + O p [ ] x + O p x ln x x ln x ln x k x = ln + n + O n= x ln x n+ <p x n [ ] k x x = n ln x n= n n + ln x lnn + + O n+ = x k lnn + xln ln x ln x nn + + O ln x n= x xln ln x = C ln x + O ln x lnn + nn n N Sn x. = e c x ln x +O xln ln x ln x. x ln x k n= n, -3 y 0 e y = + Oy n N Sn x πx = e c+o ln ln x ln x = e c + O ln ln x ln x. x : 4

12 Smarandache. x >, πx x, : lim x n N Sn x πx = e c..3 Smarandache n, Smarandache Sn m n m!, Sn = min{m : n m!, m N}. Sn, n = p α pα pα k k n, Sn = max i k {Spα i i }.. : S =, S =, S3 = 3, S4 = 4, S5 = 5, S6 = 3, S7 = 7, S8 = 4, S9 = 6, S0 = 5, S =, S = 4, S3 = 3, S4 = 7, S5 = 5, S6 = 6,., [5], Smarandache Sn,., :. k k. x 3, : Sn k kp n k ζ 3 = x 3 3 ln x + O x 3 k ln, x ζs Riemann zeta-, O k k -O..3 x 3, SMn k kp n = k ζ 3 3 x 3 ln x + O k x 3 ln x, 5

13 Smarandache SMn SM =, SMn = max i k {α i p i }, n = p α pα pα k k n..3.,..3. k n > k. i P n > n, Sn k = SMn k = kp n. ii n = mp P n, n 3 < p < P n n, Sn k = SMn k = kp n. iii n = mp n n 3 < P n n, Sn k = SMn k = kp n. Sn k., SMn k. i. n = p α pα pα r r P n n. P n > n, p kα p kα p kα r r < n k < P k n, p kα i i kp n!, i =,,, r. n k kp n!. But P k n kp n!., Sn k = SP k n = kp n..3. i. n 3 < p < P n n i ii., Sn k = kp n. n = mp n and n 3 < P n n, m < P n. Since P k n kp n!, m k kp n!. P k n kp n!, Sn k = kp n k k. x 3, Sn k kp n k x 4 3 ln x P n n 3 SMn k kp n k x 4 3 ln x. P n n 3 6

14 Smarandache n = p α pα pα r r n, Sn k = max i r {Spkα i i }. kαp = max {kα ip i }, Sn k kp ln n. i r α =, kp = kp n, Sn k kp n = 0, Sn k kp n k P n ln x P n n 3 np x p x 3 k p ln x p x p P n n 3, P n n n x p k ln x = O k x 4 3 ln x m p x m p, m m x 3. p = x 3 3m 3 ln x ln m + O m 3 x 3 ln x m. Abel [3] 4..3.,..., P n > n, Sn k = SMn k = kp n., Sn k kp n = = P n> n P n n Sn k kp n + Sn k kp n P n n Sn k kp n 7

15 Smarandache = = P n n 3 n 3 <P n n Sn k kp n + n 3 <P n n Sn k kp n + Ok x 4 3 ln x Sn k kp n. -4 n 3 < P n n, : a n = m P n m < P n. b n = m p P n m < n 3 < p < P n. c n = m P n P m n 3. b c, Sn k = kp n, 0. Sn k kp n, p p α m, α αp > P n., p < n 3..3., b x 4 3 ln x.,.3. iii Sn k kp n = + = = n 3 <P n n mp x mp 3 <p mp mp p x mp p 3 <p <p mp p mp x mp 3 <p mp Sm k p k kp m k p k Smp p kp mp p + O k x 4 3 ln x k p + O k x 4 3 ln x k p + O k x 4 3 ln x. -5 m<x 3 m <p x m.3.3 k p = k x 3 3m 3 ln x ln m + O x 3 k m 3 ln x m x 3 m <k p x m m x 3 8

16 = x 3 3 ln x = k ζ 3 3 k m e m 3 ln x x 3 ln x + O k Smarandache + O k x 3 ln x e ln x m x 3 x 3 x 3 + O m 3 k ln x ln x. -6-4, -5-6 Sn k kp n k ζ 3 = x 3 3 ln x + O x 3 k ln. x Smarandache d f n F.Smarandache d f n : d f n m n m!! d f n = min{m N : n m!!}.,,. [4], Kenichiro Kashihara, Sn d f n. d f n! Sn > d f n! d f n = min{m : m! Sn}. Sn d f n. d f n,. :.4 n, x d f n = x ln x x ln x ln ln x + O ln ln x. 9

17 Smarandache. Sn d f n d f n! Sn d f n!. i d f n ln i ln Sn [] Euler : i d f n i d f n i d f n ln i. ln i = m ln m m + Oln m, ln i = m ln m m + Oln m m ln m m + Oln m ln Sn m ln m m + Oln m, ln Sn = m ln m m + Oln m, -7 m = ln Sn ln m + O. -7 ln m ln ln Sn, m = = ln Sn ln ln Sn + O ln Sn ln ln Sn + O ln Sn ln ln Sn. -8 m = d f n -8 d f n = ln Sn ln ln Sn + O ln Sn ln. ln Sn 0

18 Smarandache ln Sn ln ln Sn ln n ln ln n. ln Sn ln ln Sn ln n ln ln n = x ln x ln ln x + O x ln x ln ln x + O x ln ln x x ln x ln ln x. -9 n, n = p α pα...pα k k n. n x n A B A [, x] α i n i =,,, k. B [, x] A n x ln Sn ln ln Sn = n x n A ln Sn ln ln Sn + n x n B ln Sn ln ln Sn. -0 Sn n A ln Sn ln ln Sn n A ln x ln ln x x ln x ln ln x O ln x ln ln x ln x ln ln x n A. - n B ln Sn ln ln Sn = np x n, p= ln Snp ln ln Snp > np x n, p= ln p ln ln p = ln p ln ln p = x ln p p ln ln p + O ln p ln ln p p x n x p x p x p = x ln p p ln ln p + O ln p. - ln ln p p x p x

19 Smarandache Abel p x ln p p ln ln p = ln x ln x ln ln x + O ln ln x -3 p x ln p ln ln p x ln ln x. -4-9, -0, -, -, -3-4 d f n = x ln x x ln x ln ln x + O ln ln x. n x..5 F.Smarandache LCM n, F.Smarandache LCM SLn k n [,,, k], [,,, k],,, k., SLn SL =, SL =, SL3 = 3, SL4 = 4, SL5 = 5, SL6 = 3, SL7 = 7, SL8 = 8, SL9 = 9, SL0 = 5, SL =, SL = 4, SL3 = 3, SL4 = 7, SL5 = 5,. SLn,,, [8] [9]., Murthy [9] n, SLn = Sn, Sn Smarandache,, Sn = min{m : n m!, m N}., Murthy [9] : SLn = Sn, Sn n? -5 [6], : -5 n n = n = p α pα pα r r p,

20 Smarandache p, p,, p r, p, α, α,, α r p > p α i i, i =,,, r. [7] : SLn = π x k ln x + c i x x ln i x + O ln k+. x i= [], ln SLn.,,.,., :.5 x >, ln SLn = x ln x + O x..5 Sn., :.6 x >, ln Sn = x ln x + Ox, Sn Smarandache.,..5. n >, n = p α pα pα s s n, α, α, α s, n. A x x, A x = ζ 3 ζ3 x ζ + 3 ζ x 3 + O x 6 exp C log 3 5 xlog log x 5, C > 0. 3

21 Smarandache [0]..5. p, k. x, ln p = x ln x + O x. pk x p, k= [3], [] [0] k x ln p p = ln x + O, ln p = x + O k x k x ln p p D. pk x p, k= ln p = p x = p x = x p x.5.. ln p ln p x ln x = D + O, ln x k x p p, k= ln p p x p x p + O x p x = x ln x + O x. ln p p + O ln p p x,. Un = ln SLn. Un. F.Smarandache LCM 4

22 Smarandache SLn n, SLn n ln SLn ln n, ln SLn ln n. [3] Un ln n = x ln x x + O ln x = x ln x + O x. -6 Un. n >, n = n, [, n] A p α pα pα s s and B. A [, n] α i i =,,, s., A [, n] square-full ; B n [, n] n / A. Un = ln SLn + n A n B.5. A n A ln SLn n A ln SLn. ln n n A = ln x A x x ln x. ln x = ln x n A B. SLn = max{p α, pα,, pα s s } [9], n B, p p n p n., SLn SLnp p. ln SLn = n B ln SLnp np x n, p= np x ln p. -7 n, p= ln SLn x ln x + O x. -8 n B 5

23 Smarandache -6-8 ln SLn = x ln x + O x Smarandache Pierced n, cn = 0 0 4n n n + Smarandache Pierced. : 0, 000, 00000, , ,. Smarandache [4] Smarandache Pierced : : : cn 0? n cn 0?, Smarandache 0 4n = n n n + = 0 4 cn 0. Smarandache cn 0.,,.,., :.7 n, cn 0. : k, n >. m >, m k n, n k, k =, n. 6

24 Smarandache. 0 mod9. : a bmodm, n : a n b n modm []. 0 4n 4 mod9, 0 4n 8 mod9, 0 4n mod9. mod9, cn 0 04n n n + mod9. s, t, cn 0 = 9s + t. t, t = 9t, cn 0 = 9s + t = 9s + 9t = 9s + t = 3 s + t, cn 0.. 7

25 Smarandache.7 Smarandache, Smarandache..7. Smarandache, Kenichiro Kashihara S x n + S x n + + S x n n ns x S x S x n.-9,,., :.8 n >, -9 x, x,, x n..9 n 3, x, x,, x n -9, x, x,, x n n..8 n 3. n =, x = x =, Sx + Sx = S + S = = 8 = SS = Sx Sx. n =,.8.,..8. n =,, -9 Sx Sx, x. n. x = x = x n =, x n = p > n, p. S =, Sp = p and Sp n = np, 8 S x n + S x n + + S x n n = n + Sp n = n + np -0

26 Smarandache ns x S x S x n = nsp = np S x n + S x n + + S x n n ns x S x S x n.- p > n, x, x,, x n =,,, p -0., -9 x, x,, x n n 3. x, x,, x n -9, x, x,, x n n. x >, x >,, x k > k n S x n + S x n + + S x n n ns x S x S x n.-3 Sn Sx i > S x n i nsx i, i =,,, k. a + a + + a k < a a a k a i > k 3, i =,,, k; k =, a + a a a, a = a = a >, a >., -3 n k + S x n + S x n + + S x n k nsx Sx Sx k.-4 k 3, -4 Sn n k + n [S x + S x + + S x k ] nsx Sx Sx k n k n + S x + S x + + S x k Sx Sx Sx k.-5 0 n k n <, -5, Sx Sx Sx k Sx + Sx + + Sx k +. 9

27 Smarandache k =, -4 n + S x n + S x n nsx Sx. -6 Sx n nsx, Sx + Sx Sx Sx x = x =, Sx > Sx >, -6. Sx = Sx =, x = x =., -6 S n 3n S n = m, m 4, n 3. Sn [ ] m [ m ] < n i., n + -8 i= i [ ] m i= i > m i= + m 4 = 3m, 4 m = S n 3n + 9 m m + = m > m.. n 3 x, x,, x n -9, x, x,, x n n n, m n m, n =, fmn = max{fm, fn}, fn Smarandache., Smarandache Sn Smarandache LCM SLn Smarandache. [4], Smarandache fn : f = ; n >, 0

28 Smarandache fn = max { i k α i + }, n = pα pα pα k k n. fn, : λ. fn = x x ln ln x + λ x + O, -8 ln x fn = 36 ζ 3 ζ3 x ln ln x + d x + O x 3, -9 ζs zeta-, d., [4], , [4],., :.0 x >, fn = x x + O.. x >, fn = 36 ζ 3 ζ3 x + O x 3, ζn zeta-.,. :.7. A. x >, n A = ζ 3 ζ3 x ζ + 3 ζ x 3 + O x 6,

29 Smarandache ζs zeta..7. B. x >, = N x 3 + O x 4, N. n B [5].., n >, fn fn = f + fn + fn, -30 n A n B A., n >, p, p n, p n. B n / A n >. fn, A.7. n B fn = O n A fn = n B x. -3 = = x + O x -30, fn = + n A = x + O x fn + n A. -3. n B fn

30 Smarandache. fn fn = 4 + n A = 4 + n A fn + n/ A fn fn. -33 A. C., n A fn = n A, fn= 3 = n A 3 + fn n C 3 = ζ 3 ζ3 x + O n C x 3 3 ζs zeta fn = ζ 3 ζ3 x + O x O n C Smarandache n, Smarandache Sn m n m!. [6], Jozsef Sandor P n : P n = min{p : n p!}, p., 3

31 Smarandache P n p n p!. P n Smarandache Sn. : P =, P =, P 3 = 3, P 4 = 5, P 5 = 5, P 6 = 3, P 7 = 7, P 8 = 5, P 9 = 7, P 0 = 5, P =,. p P p = p, n, P n = n. p, : p + P p 3p. n, [6] 4 Sn P n Sn. -35 P n,., :. x >, P n = x + O x 9..3 x >, 3 P n P n = 3 ζ x 3 ln x + O x 3 ln, x P n n, ζs zeta-.,.. x >, [, x] A B, A : p p n p > n n [, x]. B n / A. P n P n = P n = P pn = p 4 n A p n, n<p p n<p p n<p

32 Smarandache = n x n<p x n p. -36 Able 4. [7] : πx = k i= a i i =,,, k a =. n<p x n a i x x ln i x + O ln k+, x p = x x n π n πn n n= = x k n ln x + i= x n n b i x ln i n n ln i x πydy + O x n ln k+ x,-37 n x, b i. n = π 6, ln i n n i =, 3,, k n A n x n= P n = x k n ln x + b i x ln i n n ln i x i= i= + O x n ln k+ x = π x k ln x + c i x x ln i x + O ln k+, -38 x c i i =, 3,, k. B. p α, S p α α p, -36 n B P n = n B Sn n ln n x 3 ln x. -39 P n = P n + P n = π x k ln x + c i x x ln i x + O ln k+, x n A n B i= 5

33 Smarandache c i i =, 3,, k n >, P n n. [, x] A, C D, A p p > n n [, x] ; C [, x] n p n n = n p, p ; D n / A n / C n [, x]. n A, P n = P n P n P n = 0. P n P n = n A n C, P n = P p [ p +. ], x, M.N.Huxley [9] x, x + x 7. [-38] 3 p + P p p + O p n<p x n p = 3 ζ 3 x 3 ln x + O P n = p, n = n p C., -4-4 P n P n = x 3 ln x P np P np. -4 n C 3 = 3 = 3 = 3 ζ 3 n<p x p n<p x n n<p x n P p p = p + O x 3 ln x + O x 3 4 x 3 ln x 3 n<p x n p + O p 9, -43 6

34 Smarandache ζs zeta-. n D P n P n 0, P p α Sp α p ln p, P p 3 p ln p, P n P n p 3 x ln x. -44 n D 3 α ln x np α x -40, P n P n = 3 ζ x 3 ln x + O x 3 ln, x P n n, ζs zeta F.Smarandache k [5] Sn Sn P n = ζ 3 3 x x 3 + O 3 ln x ln, x P n n, ζs Riemann zeta-. [5],., k. k, n k a k n m m n k. F.Smarandache,!, [9], [8]3 [30]. [5] Sa k n,.4 k. x 3, S a k n k P n = ζ 3 3 x 3 ln x + O x 3 ln x + O k x 4 3, ln x 7

35 Smarandache ζs Riemann zeta-..5 x 3, SM a k n k P n = ζ 3 x 3 3 ln x +O x 3 ln x +O SMn SM =, n > n = p α pα pα r r SMn = max {α ip i }. i r k x 4 3, ln x n :,,.7.3 k. n, i P n > n, S a k n = SM a k n = k P n; ii n = mp P n n 3 < p < P n n, S a k n = SM a k n = k P n; iii n = mp n n 3 < P n n, k > S a k n = SM a k n = k P n; k =, S a k n = SM a k n kn 3. Sa k n., SMa k n. i. n = p α pα pα r r n, P n > n, P n = p r, α r =. Sa k P n = S P k n = k P n. S a k p α i i S p k i k pi k P n, i =,,, r. S a k n = k P n..7.3 i. ii. m p p < n 3. n = p α pα pα r r n, a k n = p β pβ pβ r r β i k, i =,,, r. n 3 < p < P n n S ak n = k P n. ii. 8

36 Smarandache n = mp n n 3 < P n n, m < P n k >, S a k P n = S P k n = k P n, S a k n = k P n. k =, a k P n =, m n 3. S a k n S P k m kn k. x 3, S a k n k P n k x 4 3 ln x P n n 3 P n n 3 SM a k n k P n k x 4 3 ln x. n = p α pα pα r r n. a k n = p β pβ pβ r r. S a k n = max i r {Spβ i i }. βp = max ip i }, S a k n i r βp kp. P n n, β = k, S a k n k P n = 0, Sa k n k P n k P n P n n 3 np x p x 3 k p p x m p x m p k k x 4 3 ln x. n x p P n n 3, P n n p, m. p x 3 = 3m 3 ln x ln m + O x 3. m 3 ln x m 9

37 Smarandache. [5] P n > n, Sa k n = SMa k n = k P n = = = = Sa k n k P n P n> n P n n P n n 3 n 3 <P n n Sa k n k P n + Sa k n k P n Sa k n k P n + Sa k n k P n + O Sa k n k P n P n n n 3 <P n n k x 4 3 ln x Sa k n k P n. -45 n n 3 < P n n, : a n = m P n m < P n. b n = m p P n m < n 3 c n = m P n P m n 3. < p < P n. b c n, n Sa k n = k P n, n a n, k > Sa k n = k P n. n 3 <P n n Sa k n k P n 30

38 = = Smarandache mp x mp 3 <p mp mp x mp 3 <p mp = m<x 3 m <p x m Sp k k p p p. -46 a n, k = Sa k n k P m k n 3, Sa n P n = = n 3 <P n n mp x mp 3 <p mp mp x mp 3 <p mp = m<x 3 m <p x m S a n psa n + p p + O p + O mp x mp 3 <p mp mp 3 + mp 3 p x , x 4 3 x 3 ln x n 3 <P n n = m x 3 m <p x m = x 3 3 ln x m e m 3 ln x Sa k n k P n p + O x 4 3 = + O m x 3 e ln x m x 3 k x 3 3m 3 ln x ln m + O x 3 m 3 ln x m x 3 + O m 3 ln x x 3 ln x 3

39 = 3 ζ 3 Smarandache x 3 ln x + O x 3 ln x Sa k n k P n = 3 3 ζ x 3 ln x +O x 3 ln x +O k x 4 3. ln x.7.5 F.Smarandache OSn [, n] Sn n ESn [, n] Sn n. [4] Kenichiro Kashihara ESn lim n OSn?,.,,., :.6 n >, ESn OSn = O. ln n [4]. : n, ESn lim n OSn = 0. 3

40 Smarandache. ESn. n >, n = p α pα pα r r n, Sn Sn = S p α i i = m p i. m =, Sn = p i, n =. M = ln n, ESn = k n Sk + + k n Sk=Sp α, α Sk M -49, kp α n αp>m, α M <p n n p + n ln n + p n α> M kp n p>m + p α n αp>m, α 3 n p α + kp α n αp>m, α 3 n p α p n p> M, α 3 n ln n + n p α + M <p n p n αp>m, α p kp α n αp>m, α k n p + n p α p α n αp>m, α 3 p n αp>m, 3 α<p k n p α n ln n + n + n M M n ln n , [ ]. M p M, αp = p, αp M p. u = p αp. Sk M k, Sk = Sp α, p M Sk p α M!,, α j= [ ] M p j n p α M p. Sk M k u k u, du. = + αp = [ ] M + p Sk M d u p M p M 33

41 Smarandache = exp p M [ ] M ln +, -5 p expy = e y. [3] [] πm = p M = M M ln M + O ln M ln p = M + O : M ln M p M ] p M = p M [ M ln + p p M [ ln p + M ln p ln πm lnm p M = M lnm ln M ln + M p p ln p + p M M M + O = O ln M M = ln n, -5-5 : Sk M exp p ] M. -5 ln M c ln n, -53 ln ln n c. exp c ln n ln ln n n ln n, -49, : ESn = k n Sk n = O. ln n OSn + ESn = n, 34 n OSn = n ESn = n + O. ln n ESn OSn = O n ln n n + O n = O. ln n ln n

42 Smarandache.6. 35

43 Smarandache Smarandache. Smarandache Smarandache :. n >, Smarandache Zn m { [ } + + m] n., Zn = min m : m N : n mm+. Smarandache,,, :. n, Smarandache Zn. :. : n =, Zn =.. n, Zn < n. : :Z = 3, Z4 = 7, Z8 = 5..3 p 3, Zp = p. : Zp = m, m. m k = k= mm +, m mm + p., p m m +, p = m + p = m, p., p =, Z = 3..4 p 3 k N, Zp k = p k. p =, Z k = k+. 36

44 Smarandache : Zp k = m, m. m k = k= mm +, m p k mm +., p k m m +, p k = m + p k = m, p., p =, Z = 3..5 n, Zn = max{zm, m n}. : n, : Zn max{zm, m n}. Zn = p, Zm = q, m n. q > p, n pp + qq +, m..6 Zn, Zm + n Zn + Zm. Zn, Zm n Zn Zm. : : Z + 3 = Z5 = 4 5 = Z3 + Z,.7 Z 3 = Z6 = 3 3 = Z Z3. n lim n Zk. k= :, Zn n k= Zk > n p=3 Zp = 3 p n p > 3 p n p., p n., p k= Zk. 37

45 Smarandache.8 lim n n k= Zk k. :, Zn n k= Zk k > n p=3 Zp p = 3 p n p p > 3 p n p. 3 p n n., p k= Zk k..9 m, n, Zn = m.. Smarandache [4] [44], Kenichiro Kashihara David Gorski Zn,. : p 3, Zp = p ; p 3 k N, Zp k = p k ; k N, Z k = k+ ; k > 0, n k, Zn < n., Kenichiro Kashihara [4] and M.L.Perez [3] Zn, : A Zn = Zn +. B r, s Zs Zs + r. A, Kenichiro Kashihara,. n 60. B, Kenichiro Kashihara :,. Zs Zs +, Z Z3 = 3 = ; x 60 5, Z3 Z33 = 63 = 5.,,. 38.

46 Smarandache. Zn = Zn +. m, Zn = Zn + = m, Zn, mm + mm + n, n +, nn + mm +, n < m. - m = Zn n n, m = Zn + n n, m n r s Zs Zs + r. α N, s = α, Zs = Z α = α+, Zs + < s = α. Zs Zs + > α+ α = α. α > r, Zs Zs + > r 39

47 Smarandache α > log r+. s = α = r + Zs Zs + r...3 x >, ZSn = π x x ln x + O. ln x P n n. A and B. A = {n n x, P n n}, B = {n n x, P n > n}. ZSn = ZSn + ZSn -3 n A n B n A ZSn = n B ZSn n A P n> n Z n ZP n = n p x n n ln n x 3 ln x -4 p = π x x ln x + O.-5 ln x 3, 4, Smarandache Smarandache 40

48 Smarandache.3. Smarandache Zn + S n = n -6,. :.4 n = 6; n n p. n = p k, p 3, k..,.3. n, Zn = n n p n = p α, p, α. Zn = n, Zn n nn, n, n =, n. n. n = u v, u, v = u > v >. k = u v u, u x u mod v, u v. n = u v k k = u v u u v u+. Zn Zn k = u v u < n, Zn = n. n, n, n p.. S n = m. n =. n >. Smarandache Zn Zn Zn + + m Zn + = n Zn +. Zn n m Zn +, m! n, 4

49 Smarandache m Zn + = kn Zn = kn m. kn m + m = n. -7 S n = m m! m n mm. m! m m =, 3. m =, k =, Zn = n. Zn = n n p, n = p α, p, α. p p α. m =, n > n mm, n =, n =. m = 3, mm = 6, n mm = 6 m! = 6 n n = 6. n = 6., n = 6; n n p...3. Smarandache n, F.Smarandache Zn m n mm+. Zn = min{m : m mm+ N, n }. Zn Zn : Z =, Z = 3, Z3 =, Z4 = 7, Z5 = 4, Z6 = 3, Z7 = 6, Z8 = 5, Z9 = 8, Z0 = 4, Z = 0, Z = 8, Z3 =, Z4 = 7, Z5 = 5, Z6 = 3,. Zn,,. [] [44] [4] [46] [5]. [5] Zn + = Sn Zn = Sn,, Sn Smarandache., J.Sandor [3] Zn, Zn Z n : Z n m mm+ n. mm+ Z n = max{m : m N, n}, N. Z n,, [3] 4

50 Smarandache Z n, Zn Z n : Z n 8n + Zn. -8 [5], J.Sandor p 5 n Z p n =, Z 3 n =. Smarandache,, : Zn + Z n = n -9 : A., n = 6; B. p 5. [5] B. A,., : Zn + Z n = n -0.5 n, n 3 n. n = k, k. [5],. A.. Smarandache Zn k, Z k = k+. Z k =. n = k 3. Z k + Z k = k+ + = k+. 43

51 Smarandache 3 n =,. n 3, n n n, Zn Zn n. Zn + Z n n + 8n + < n. n 3., 3 n = k k,., n 3, n k, n = k h, h 3. Zn = m. m, Zn n mm+. n, m/ = u, n = u v, m = u m. u, v = v m +. u > v v m + u m + 0 mod v, m u mod v m = v u, u v u u mod v. m = u v u. u v, u > v : m = u v u u v. Zn + Z n u v + 8uv + < uv u + u < n. n 3. u < v, v m + m = hv. u m : hv 0 mod u. h = v, v v mod u. v u, u < v : Zn + Z n v v + 8uv + u v + v < n. v = u, v mod u, v = ut. u < v 8uv + Zn + Z n v v + 44

52 Smarandache < u v + v + = uv = n. m+ m. m +. n, = u, n = u v, m + = u m. v m. m = v h. u > v, u m mod v. m = u, u u mod v. m = u m = u u u v. 8uv + Zn + Z n u v + u v u + u < uv = n. v > u, v h + 0 mod u. h = u v. m = v h = v u v. v u, m = v u v v u, 8uv + Zn + Z n v u + u v v + v < n. v =, v = tu + u = v t. m = v u, 8uv + Zn + Z n v u + < u v v + v = n. n 3 n = k, k Smarandache Smarandache Zn, [] [] [3] [3]. a. α p, Z p α = p α ; b. α, Z α = α+ ; c. Zn, Zm + n = Zm + Zn ; 45

53 Smarandache d. Zn, Zm n = Zm Zn. Un : U =. n > n = p α pα pα s s n, : Un = max{α p, α p,, α s p s } Smarandache. fn, f =, n > n = p α pα pα s s n, fn = max {f p α, f pα,, f pα s s } Smarandache.,,,, [9] [44]. [9] : Un P n = ζ 3 x 3 3 ln x + O x 3 ln x ζs Riemann zeta-, P n n. Zn = Un Zn+ = Un,, :.6 n >, Zn = Un n = p m, p, m p+. m p+ m >..7 n Zn + = Un n = p m, p m p. m p. 46

54 Smarandache Zn = Un Zn + = Un., [, 00], Zn = Un 9, n =, 6, 4, 5,, 8, 33, 66, 9. Zn + = Un [, 50] 9 n = 3, 5, 7, 0,, 3, 7, 9,, 3, 6, 9, 3, 34, 37, 39, 4, 43, n =, Zn = Un =. n =, 3, 4, 5, n Zn = Un. n 6 Zn = Un, n = p α pα pα s s n, Un = U p α = αp. Zn Un αp n : n αpαp +, p α n - α =. α >, p α n p α αpαp + - p, αp + = p α α. p, p α > α, p α α. p = α = = 0! α = p. n = p m. p m pp+ m p+,. m. n = p, Zp = p, Up = p Zn = Zn = p, Un n = p m, m p+ Un = p Zn = Un. n > Zn = Un n = p m, m p n = Zn + = Un. n > Zn + = Un, Un = U p α = αp. Zn + = Un Zn = αp. Zn Un n αpαp, p α n -3 47

55 Smarandache p, αp = 3 p α α. α = p. n = p m. 3 m p p. n = p m, m, n Zn + = Un. Zn + = Un n = p m, m p Smarandache.4. Smarandache [4], Kenichiro Kashihara n Smarandache Zn n.,, n, Kenichiro Kashihara, Z4 = 7 4 Z3 = 3., n Zn n?, :.8 n, Smarandache Zn n n =, 3, 4.. n >, a a, n =. Euler a φn modn, φn Euler, φn n n. a, n, m a m modn a m modn m. m = φn, a n., n., :.4. m >, m m =, 4, p α, p α, p, α. 48

56 Smarandache [3] [] 4.. Z = 3 Z4 = 7 4. n = p α, p. Zn n = p α, Zn Zn = p α, p α n = p α Z n = p α mod p α p α n = p α, = φ p α = p α p. α =, p =. n = + = 3. n = p α p, Zn n n = p = 3. n = p α, Zn C. p α 3 mod 4, Zn Zn = p α. p α, p α = p α >, Zn = p α n = p α.. p α mod 4, Zn Zn = p α. p α, p α = >, Zn = p α n = p α. n Zn n n =, 3,

57 Smarandache p, p, p < p, p p =,., 3 5, 5 7, 3, 7 9, 9 3,,.,, [58], [], [0] [3]. [4], Kenichiro kashihara. :. 3. p, p p + p! + p + p +! + p Kenichiro kashihara [4] : : p, p p + { p! p + } + p + p + p +. :,,,.,., : 50

58 3. p, p p + { p! p + } + p + p + p p =, p + = 3.,. p p +, { p! p + } + p + p + p +., p, p! modp. p p! +. p! + p. p +, p +! + 0modp +.. { p! p + } p + p! p p + + 0modp + p! + 0modp +. p! + p + + p + p + = p! + p + p! +, p + 5

59 Smarandache { p! p + } + p + p + p +. p! { p + } + p + p + p + 3-, p p +., : a p p + ; b p, p + ; c p, p +. a, a b, c d p = a b, p + = c d., a < p, b < p, c < p +, d < p +. p = 4, p + = 6, 3. p > 4, a p! b p!, p = ab p! a = b, a p!, p p!. p! p.. p +! p + p + p + { p! p + } + p + p + p +. b, p! + p 5

60 ,. { p! p + } p +. c,,,. p! + p + p + p +! p + p + = p! + p p! p p +! + p + p { p + } + p + p + p = p! p + + p +! p + p +! + p + + p +. + p = p! + p +! + + = p p + p! + p +! + p p + + p + p! p = p! p p +! p + p + p +! p + + p + p +. 53

61 Smarandache p, p +, p! + p + p +! p + + p +, p +. p + p >, p. p! p + p +! + p + + p, p p + p >, p! p p + p p +! p + + p + p +, 54

62 k k 4. k k 4. k, n >. m > m k n, n k, k =, 3, n. 4. k k,.,,. [55] [56]. H.L.Montgomery R.C.Vaughan [57] k, Q k x x k, Q k x = x ζk + O x +ɛ k+, ɛ ζk ζ-. Q x = 6 π x + O x 9 +ɛ 8., a n k n k. [4], Mladem Krassimir a n, a n < [ 4 n + 3n + 4], [x] x. c > a n < cn? c < 55

63 Smarandache [5], Xigeng Chen. : a n <.8n., [59], : 4. : k. n >, a n k = ζkn + O n k, ζk = n= n k ζ-. : 4. ε > 0, N > 0 n > N ζk εn < a n k < ζk + εn..64 εn < a n <.65 + εn. 4. k 3, : a n k lim n n a n = ζk; lim n n = π 6.. Q k x x k. Q k a n k = n., a a = b k d, b, d k. Möbius : Q k x = µd = µd = µd d k x d k n nd k x /d k 56

64 k = d k x = x d= x µd d k + O = x µd d k + O x k d k x d= µd d k + O x k = x ζk + O x k, µd d k = ζk. Q k a m k = m, x = a m k, m = ζk a mk + O a k m k = ζk a mk + O m k : a n k = ζkn + O n k.. ζ = π, c c > π

65 Smarandache Smarandache 5. Smarandache 5. n, Smarandache Zwn m n m n. Zwn = min{m : m N, n m n }. F.Smarandache Only Problems, Not Solutions, : 5. Zwn. 5.. n, n = p α pα pα r r n, Zwn = p p p r., Zwp = p, p. 5.. n, Zwn = n n, Zwn n n= Zwn n Zwn,, GCDm, n =, Zwm n = Zwm Zwn. 58

66 Smarandache 5..6 Zwn,, Zwm + n Zwm + Zwn n, Zwn α > 0 x, Zwn α ζα + xα+ [ ] = ζα + p α + O x α+ +ɛ. p + p n, 0 < Zwn n. ɛ > 0, n Zwn n < ɛ Zwn n =,. n, Zwn ; n, Zwn Zwn = Zwn +. n, n k= Zwk > 6 n π n= Zwn a, a R, a > 0. 59

67 5..6 n= Smarandache α, s s α > α > 0, Zwn α n s = ζsζs α ζs α p [ ] p s + p α, ζs Riemann zeta-, p. 5.3 Smarandache Smarandache,,, Zwn [60], : Zwn. n= lim k Zwk θk, θk = ln Zwn. [6]. Zwn n= Zwn. lim k θk, θk = lnzwn,. : 5. Zwk θk, Zwk lim k θk = 0 θk = ln Zwn,. 60

68 Smarandache 5.3. k, n k µn = 6k π + O k, µn, µn = d n µd, [3]. µn = µd n k n k d n = µd md k = d k = k d k µd = 6k π + O k m k d µd d + O k d= µd d + O k,. θk = lnzwn µn lnn µn ln k n k n k k = ln k k n k µn = ln k µn µn n k n k 5.3. θk 6k ln k π + O k 6

69 Smarandache Zwn lim k θk lim k Zwn ln k 6k π + O lim k k k ln k 6k π + O = 0 k Zwn 5. Scientia Magna,,, 5. n >, n lnzwk = + O. n ln k ln n k= Un = n k= lnzwk. Un. ln k k > Zwk Zwk k, k Zwk k lnzwk ln k, : Un = n k= lnzwk ln k n k= ln k ln k = n n. 5- Un. k n, k = p α pα pα s s k, [, n] A B, A [, n] α i i =,,, s k ; B [, n] α i = i s k. 6 Un = n k= lnzwk ln k n k= lnzwk ln n

70 Smarandache = ln n k A lnzwk + ln n lnzwk. 5- A A [, n] Square-full, : lnzwk ln k n ln n. 5-3 k A k A, n B, p, p n p, n =. p : ln p = ln n + O, p k n k n k n n ln p = n + O ln n ln p p k B = D + O, ln n D. lnzwk = lnzwpk = k B pk n p, k= ln p = pk n p n p, k= = p n = n p n ln p ln p p ln p n p n p + O n p n ln p p pk n p, k= k n p p, k= ln p + lnzwk + O ln p p n = n ln n + On , : Un = n k= lnzwk ln k ln n lnzwk + O n ln n k B 63

71 Smarandache ln n n ln n + On + O n ln n n = n ln n : n n k= lnzwk ln k = + O. ln n. n, 5. n, lim n n n k= lnzwk ln k = Zwk θk 5.3 k >, θk = n k ln Zwn, Zwk θk = Zwk ln Zwn = O n k. ln k, x > n, Möbius µn : µn = µd d n 64 n= µn = µd d n µn n = ζ = 6 π

72 Smarandache = md x = d x = d x = x = x µd µd m x d x µd d + O d= µd d 6 π + O = 6 π x + O x. d> x µd d x + O x + O µd d x. Zwn = n, n, θk = ln Zwn n k n k µn ln n k n k µn ln k = ln k k n k µn = ln k µn µn. 5-6 n k n k θk ln k µn µn n k n k 6 ln k π k + O k = 3 π k ln k + O k ln k

73 Smarandache Zwn n, 0 < Zwk θk k 3 π k ln k + O k ln k = O ln k Zwk θk. : = O. ln k 5. k, Zwk lim k θk = Smarandache Smarandache, n >, Zwn = p, p n p n n []. Zwn., Zw =, Zw =, Zw3 = 3, Zw4 =, Zw5 = 5, Zw6 = 6, Zw7 = 7, Zw8 =, Zw9 = 3, Zw0 = 0,. n, Zwn = n. [60], Felice Russo Zwn, : : Zwn = Zwn + Zwn + n. : Zwn Zwn + = Zwn +. 66

74 Smarandache 3: Zwn Zwn + = Zwn + Zwn : Sn = Zwn n, Sn Smarandache.,., : 5.4. Zwn = Zwn + Zwn + Zwn Zwn + = Zwn + Zwn Zwn + = Zwn + Zwn n Sn = Zwn, Sn Sn = min {k : k N, n k!} Smarandache., 5.4. Zwn = Zwn + Zwn +. I n =. n =, = Zw 3 = Zw Zw3. n >, n = n 0, Zwn 0 = Zwn 0 + Zwn 0 +. n = n 0 p, p Zwn 0. Zwn 0 = Zwn 0 + Zwn 0 +, p Zwn 0 + Zwn 0 +., p Zwn 0 + or p Zwn 0 +. a p Zwn 0 +, p n 0 +, p n 0 p n 0 + n 0 =,. b p Zwn 0 +, p n 0 +, p n 0 p n 0 + n 0 =, n 0 = p =. 67

75 Smarandache = Zw 3 = Zw3 Zw4,.., , Sn : 5.4. n = p α pα pα k k Sn = max i k {Spα i i }. n, [64] p, Sp k kp; k < p, Sp k = kp, k. [65] p Sn = Zwn. n Sn = Zwn. n Sn = Zwn, n = p p p k p α k k, p i, p k > α k = p p p k., Sn Zwn Sn = p p p k p k Zwn = p p p k p k. n = p p p k p α k k p i, p k > α k = p p p k Sn = Zwn. k, n Sn = Zwn

76 Smarandache Smarandache 6.,. Smarandache. 6. F.Smarandache LCM F.Smarandache LCM : 6.. n, F.Smarandache LCM SLn k n [,,, k], [,,, k],,, k., SLn SL =, SL =, SL3 = 3, SL4 = 4, SL5 = 5, SL6 = 3, SL7 = 7, SL8 = 8, SL9 = 9, SL0 = 5, SL =, SL = 4, SL3 = 3, SL4 = 7, SL5 = 5, SL6 = 6,. SLn n = p α pα pα r r n, SLn = max{p α, pα,, pα r r }. 6- SLn,,, [9], [6], [7] [66]. Murthy [9] n, SLn = Sn, Sn F.Smarandache., Sn = min{m : n m!, m N}. Murthy [9] : SLn = Sn, Sn n? 6- Le Maohua [6], : 69

77 Smarandache n = n = p α pα pα r r p, p, p,, p r, p α, α,, α r p > p α i i, i =,,, r., Lv Zhongtian [7] SLn, k x > SLn = π x k ln x + c i x x ln i x + O ln k+, x i= c i i =, 3,, k. [66] [SLn Sn], : [SLn Sn] = 3 3 ζ x 3 k i= c i ln i x + O x 3 ln k+ x ζs Riemann zeta-, c i i =,,, k. F.Smarandache LCM SLn Dirichlet dn,. : 6. k. x, : dn SLn = π4 36 x k ln x + c i x x ln i x + O ln k+, x dn Dirichlet, n dn =, c i i =, 3,, k. d n,. dn SLn i=,

78 Smarandache, n x U V, U p p n p > n n; V [, x] U. dn SLn = n U p n, n<p = np x n<p dn SLn = np x n<p p dn = n x dn dnp SLnp n<p x n p. 6-4 πx = p x. Abel [3] 4. [67] 3.: πx = k i= c i i =,,, k c =. n<p x n c i x x ln i x + O ln k+, x p = x x n π n πn n = a i. n U x k n ln x + n= i= n= dn SLn = x ln x x n n a i x ln i n n ln i x n = π 6 πydy + O dn n = n = π4 36, n= n x dn n + n x i= x n ln k+ x k a i x ln i n n ln i x, 6-5 x + O ln k+ x 7

79 Smarandache = π4 36 x k ln x + b i x x ln i x + O ln k+, 6-6 x i= b i. V, V n V, n n = p α pα r r, SLn = p r n SLn = max α i. : dn SLn n np x i r {pα i i } = p α i i, dn np + np α x α α + dn p α dn n ln n x 3 ln x, 6-7 : dn = x ln x + Ox. U V 6-3, dn SLn dn SLn = dn SLn + n U n V = π4 36 x k ln x + b i x x ln i x + O ln k+, x i= b i i =, 3,, k Smarandache Smarandache S n m m! n, n, 7 S n = max{m : m! n, m N}.

80 Smarandache,,., [6], J.Sandor, k k + q : S k!k +! = q, [68]. [69], S n, S n. α, n α, α >. n= S n n α = ζα n= n! α, ζα Riemann zeta -. S n : { S max{m : m!! n, m N}, n; n = max{m : m!! n, m N}, n. S n., S n, n s n= n=,. : 6. s >,, n= S n n s n= = ζs s + ζs Riemann zeta-. S n n s m= : m +!! s +ζs m= m!! s, 73

81 Smarandache 6. s =, 4, i n= S n n = π 4 m= m +!! + π 3 m= m!! + π 8. ii n= S n n 4 = π4 48 m= m +!! 4 + π4 45 m= m!! 4 + π4 96.,. s >, S n << ln n, n= S n n s = n= n S n n s + n= n S n n s. n= S n n s S n n, S n = m, m!! n. n = m!! n m + n. s >, n= n S n n s = = = = m= m= m= n= n S n=m m m!! s m n s = n= n m+ n m m!! s n= n n s m= n= n m= n s n= n m+ n n s n= n m m!! s m= m m!! s n s m + s n s m m +!! s 74

82 = = Smarandache + n s n= n = ζs n s n= n + m= m= s + m + m +!! s m +!! s m= m= m m +!! s m +!! s. 6-8 n, S n = m, m!! n. n = m!! n m + n. s >, n= n S n n s = = m= m= n= n S n=m m m!! s m= m n s = n= m+ n m= n= m+ n n s m m!! s n s m = m!! s n s m + s n s m= n= n= m = n s m!! s m m +!! s n= m= m= = n s + m +!! s n= m= = ζs + m +!! s = ζs m= m!! s

83 Smarandache n= S n n s = n= n = ζs S n n s + n= n s +. S n n s m= s =, 4, ζ = π π4 6, ζ4 = 90 : n= S n n = ζ 3 4 = π + 8 = π 4 m= + m= m +!! s m +!! + ζ m= m +!! + π 3 m +!! + π 3 m= + ζs m= [3] m= m= m!! m!! + π 8 m!! m!! s. n= S n n 4 = π4 48 m= m +!! 4 + π4 45 m= m!! 4 + π n, Ωn Ω = 0, n > n = p α pα pα k k n, Ωn = α p +α p + α k p k. Ω =, Ω3 = 3, Ω4 = 4, Ω5 = 5, Ω6 = 5, Ω7 = 7, Ω8 = 6, Ω9 = 6, Ω0 = 7,. n Ωn,! m n, Ωn : Ωm n = Ωm + Ωn. 76

84 Smarandache,,.,,, Ωn!, Ωn,., : 6.3 k. x >, Ωn = x k i= c i x ln i x + O ln k+, x c i i =,,, k c = π. 6.4 s >, n= Ωn n s = ζs p p p s, p s p, ζs Riemann zeta -. s = 4 ζ4 = π n= Ωn n 4 = π4 90 p p p 3. p 4, : 6.4. x >, Ωn = x ln ln x + C + O ln x, C. 77

85 Smarandache, [70] [7] [, x] A B, A [, x] P n > n n, P n n ; B [, x] P n n n. A B Ωn Ωn P n Ωn Ωn = Ωn + n A n B = Ωpn + O Ωn P n p n<p n B = p + Ωn + O p n<p = n x n<p x n p + n x = p + π n x p x n n x n<p x n Ωn n x n Ωn + O πx x. k, [67] 3. πx = x k i= a i i =,,, k a =. 6- n x x π Ωn n x Ωn + O Ωn x 3 ln ln x, 6-0 a i x ln i x + O ln k+, 6- x x ln x n x 6- Abel [3] 4. p = x πx p x 78 x 3 πydy Ωn n x 3. 6-

86 = x = x k i= k i= Smarandache a i ln i x x 3 b i ln i x + O b i i =,,, k b =. 6-3 ζ = π 6 p = [ x n n x p x n n x = x k a i y x y ln i y + O ln k+ dy + O y ln k+ x i= x ln k+, 6-3 x k i= k i= b i ln i x n x ] + O n ln k+ x c i x ln i x + O ln k+, 6-4 x c i i =,,, k c = π. 6-0, 6-, Ωn = x. k i= c i x ln i x + O ln k+. x 6.4. Ωn fs = n s, Ωn n, n= s > fs fs = = n= p = ζs Ωn n s = Ωp n p s n s = p + Ωn p s n s p n= p n= n s + p s p s p p s n= + fs p p p s n= Ωn n s

87 Smarandache 6-5 n= Ωn n s = ζs p p p s, p s p, ζs Riemann zeta Smarandache n, Smarandache P p n n., P p n P p =, P p =, P p 3 = 3, P p 4 = 5, P p 5 = 5, P p 6 = 7, P p 7 = 7, P P 8 =, P p 9 =, P p 0 =, P p =, P p = 3, P p 3 = 3, P p 4 = 7, P p 5 = 7,. n, Smarandache p p n n. p p =, p p 3 = 3, p p 4 = 3, p p 5 = 5, p p 6 = 5, p p 7 = 7, p p 8 = 7, p p 9 = 7, p p 0 = 7, p p =,. Only problems, Not solutions [], 39, F.Smarandache {P p n} {p p n}.,,., Smarandache. I n = {p p + p p p p n}/n I n = {P p + P p P p n}/n. [4] 0, Kenichiro Kashihara : A. lim n S n I n.,. 80 B. S n lim n I n.,.

88 Smarandache A,. B,.., S n I n,., : 6.5 n >, S n = + O n 3. I n : 6.3 n S n /I n, S n lim =. n I n [4] B.,. : 6.5. x >, p n+ x p n+ p n x 3 8 +ε, p n n, ε. D.R.Heath Brown [7] [73] x, x x + x 3 P. x, P n P n x P n P n x 3 8 +ε 8

89 Smarandache P n P n x 3. P x x + x x >, P p n = x + O x 5 3 p p n = x + O x 5 3.,. P k k. P p n P r, P r+ P r, n P p n = P r. P p n = = P n+ x P n+ x P n P n+ P n = P x P n+ P n P n+ x P n+ x P n+ P n P n+ P n, 6-6 P x P x x P x = x + O x , 6-7 P p n = x + O x O x 3 +ε 8 = x + O x

90 Smarandache 6.5., p p n = P n x = P n x = P x + P n P n P n P n Pn + P n P n p n x P n P n P n x,. n >, I n S n I n = {p p + p p p p n}/n = [ ] n n + O n 5 3 = n + O n S n = {P p + P p P p n}/n = [ ] n n + O n 5 3 = n + O n S n I n = n + O n 3 n + O n 3 = + O. n. n ,,,,! 83

91 Smarandache,! [4], Kenichiro Kashihara x y + y z + z x = 0 6-0,!, Kenichiro Kashihara [4] : 6-0 x, y z, 6-0!, 6-0 [4]!,. : 6.6 x y + y z + z x = 0, x, y, z =,,,,,,,,. [4 ]! 6-0, n : x x + xx xx n n + xx n = 0!. : a. 6-0 x, y, z, x 0, y 0, z 0. x, y, z, x, y, z, x, y, z, 0 0 =, 0, x, y, z 6-0. b. x, y, z, x, y, z. x, y, z 6-0, a x, y z x y + y z + z x, x y + y z + z x > 0, x, y, z 6-0.! c. x, y, z 6-0, x, y, z. x, y, z 6-0 x, y, z, 84

92 Smarandache a x, y z. x, y z x y + y z + z x > 0, x, y, z 6-0! x, y z, 6-0 x y y + = 0 z z x y z z x + x y y z = x y z x. x = s t, y = s t, M = z x, s i, t i, i =,. M. : s t z M + s t s t s t z = s t s t M. s s t = s z. s s t = s z. z, s s,! x, y z. x, y, z,, x > 0, y > 0, z < 0 x > 0, y < 0, z > 0., 6-0 x y + y + z z x = 0.,! x > 0, y < 0, z > x y y z + z x = 0 = x y y z x y z x.,,! x < 0, y > 0, z > 0 : x y + y z + z x > 0, 6-0! 85

93 Smarandache x, y, z,, x > 0, y < 0, z < x y y + z z x > 0 y z x y + y z x y z x = 0. y z x y x y y z. x y = y z y z x y z x = 0! x < 0, y > 0, z < 0. x, y, z, x y + y z + z x > 0, 6-0. x < 0, y < 0, z > 0, 6-0 x y y z + = 0 z x z x + x y = y z z x x y.,,! x, y, z 6-0, x, y, z. d. x, y, z 6-0, x, y, z. x, y, z 6-0 x, y, z, x, y z x y + y z + z x < 0 x, y z x y + y z + z x > 0 ; x, y, z x y, z 6-0 y z x y + = z x y z., ; x z, y, 6-0 x y y z + z x = 0 = x y y z z x x y.,,! 86

94 Smarandache y z, x, 6-0 x y + y z + z x = 0 = x y z x z x y z.,,! x, y, z x, y z, 6-0 z x y z x y = y z x y.,,! y, x z, 6-0 z x y z x y = z x y z.,,! z, x y, 6-0 z x y z x y = x y z x.,,! x, y, z 6-0, x, y z. e., x, y, z 6-0, x, y z,. x y z. x, y z, x y + y z + z x > 0. x, y z, 6-0 x y y z = y z z x + z x x y.,,! x, y, z 6-0, x, y z. x, y z, 6-0 z x y z x y = x y z x z x y z.,,! 87

95 Smarandache y, x z, 6-0 x y y z z x = z x + y z. x,, z, x y, 6-0 x y y z z x + z x = x y.,,! x, y, z 6-0, x, y z. x, y z x < 0, y > 0, z > y z z x + = x y z x. z x =, z =, y + = x y. x < 0, y + = x y x =, y =. x, y, z =,, 6-0. x > 0, y < 0, z > 0, 6-0 = x y y z z x.,,! x > 0, y > 0, z < 0, x, 6-0,! x y + y z + z x = 0., x, y z 6-0, x, y, z =,,,,,,,,.. 88

96 Smarandache 6.7 k 0. n >, n n = p α p α p α s s, Ωk, n : Ωk, = 0, Ωk, n = α p k + α p k + + α s p k s. Ω, = Ω, 3 = 3, Ω, 4 = 4, Ω, 5 = 5, Ω, 6 = 5 Ω, 7 = 7, Ω, 8 = 6, Ω, 9 = 6 Ω, 0 = 7,., m n : Ωk, mn = Ωk, m + Ωk, n. 6- Ωk, n : Ωk, d = dn Ωk, n, d n dn Dirichlet, d n n. k = 0, Ω0, n Ωn: n,. Ωk, n Ωn! Ωn,, [56] [7] x >, : x Ωn = x ln ln x + C x + O, ln x C. Ωk, n,. : 6.7 k > 0. x 3, Ωk, n P k n = ζk + 4 k xk+ ln x + O x k+ ln 3 x P n n, ζm Riemann zeta-. ζ = π 6, : 89,

97 Smarandache 6.4 x 3, [ Ω, n ] P n = 4π 3 x 3 ln x + O x 3 ln 3 x.. [, x] n : A, B C. A n [, x ] P n > n n ; B n [, x ] n 3 < P n n n ; C n [, x ] P n n 3 n. A, B C Ωk, n P k n = Ωk, n P k n + + n B n A Ωk, n P k n + n C Ωk, n P k n n A, P n > n, n, n = m P n, m < P n. Ωk, n n k ln n P n n, Ωk, n n ln n, Ωk, n P k n = Ωk, mp p k 90 n A = mp x = = m<p Ω k, m = m x P m> m<p x m m mp x p >m = mp <p m x 4 m<p x = m x 4 m m<p x m mp x m<p m x m<p x m Ω k, m + Ω k, m Ω k, mp + O x mp mp <p mp <p m x P m m<p x m m m x Ω k, m m k ln m p x m Ωk, m + Ωk, p + O x mp x mp x k+ ln x Ω k, m + p k Ωk, m + p k

98 Smarandache + O x k+ ln x Abel [3] 4. [67] 3. πx = = x x ln x + O ln x p x : m x 4 = m x 4 = x m<p x m m<p x m m x 4 = xk+ ln x p x m m x 4 p k mp <p x mp p k [ p k m ln x mp m k+ + O = ζk + k xk+ ln x + O x mp ln x mp x k+ + O ln 3 x x k+ ln 3 x x k+ ln 3 x ] x + O mp ln x, 6-4 ζm Riemann zeta-. m x 4 m<p x m mp <p Ω k, m = O x mp x k+ ln x 6-5 m x 4 m<p x m mp <p p k Ωk, m = O x mp x k+ ln x , 6-4, n A Ωk, n P k n = ζk + k xk+ ln x + O x k+ ln 3 x

99 Smarandache n C, P n < n 3, Ωk, n P k n n C n k 3 x k 3 + xk+ ln 3 x. 6-8 n B, n : n = m p, m < p; n = m p p, m < p < p. Ωk, n P k n n B = m p x m<p = m p x m<p Ωk, mp p k + Ωk, m + p k m p x m<p Ωk, m + p k m x Ωk, m + p k m p p x m<p <p m p p x m<p <p p x m Ωk, mp p p k Ωk, m + p k. 6-9 p k xk+ ln x xk+ ln 3 x m p p x m<p <p = m x 3 m<p x m p <p x p m Ω k, m + p k Ωk, m + p k m x 3 m x 3 m<p x m m<p x m x k+3 3 ln ln x ln x p <p p <p x p m x p m Ω k, m m k xk+ ln 3 x. 6-3

100 Smarandache m x 3 m x 3 m<p x m m<p x m p <p p <p x p m x p m p k Ωk, m p k m k x 5k+6 6 ln x xk+ ln 3 x , Abel m x 3 = m x 3 = m x 3 m<p x m m<p x m m<p x m p <p p k [ x p m p k x mp ln x mp p k x m ln x + O mp x k+ = ζk + k xk+ ln x + O n B ln 3 x ] x + O mp ln x x k+ ln 3 x Ωk, n P k n = ζk + k xk+ ln x + O x k+ ln 3 x 6-, 6-7, : Ωk, n P k n 4 = ζk + k xk+ x k+ ln x + O ln 3, x ζn Riemann zeta Smarandache n, Smarandache SP ACn k n + k. SP AC = 93

101 Smarandache, SP AC = 0, SP AC3 = 0, SP AC4 =, SP AC5 = 0, SP AC6 =, SP AC7 = 0, SP AC8 = 3, SP AC9 =,. [],, F.Smarandache {SP ACn}. : A n = n n SP ACa a= [4] Kenichiro Kashihara lim n A n = n n SP ACa a= {A n }!,, {A n }, [4]! 6.8 n, A n = n SP ACa ln n + O. n a= {A n }, [4]., n, n [n, n + n 7 ]. p q n n 7 p n n < q n + n 7. 94

102 Smarandache : [76] [77] π x x, π x = x x ln x + O ln. x : [67] [].. n, = p < p < p 3 < < p m n [, n]. SP ACa p i, p i+ ] a p i <a p i+ SP ACa = p i+ p i + p i+ p i = p i+ p i p i+ p i. SP AC =, SP ACa = + a n : = = p i+ n p i+ n p i+ n p i+ n SP ACa + p i <a p i+ p i+ p i p i+ p i p i+ p i p i+ n p m <a n p i+ p i SP ACa p i+ p i p m p m = p i+ p i p i+ p i p i+ n p i+ n p i+ n 95

103 Smarandache = p i+ n p i+ p i πn. : p i+ n p i+ p i p m. πn 6-38 A n : na n p m πn p m = [ p pm m πn A n [ ] n p pm m. πn ] n p m n 7 A n [ n + O ] n 7 n [ n ln n + O n ln n = ln n + O. pm ] + O = n n + O n ln n + O n 9 n ln n + O [ ] lim ln n + O = +, n A n.. lim A n = +. n 6.9 F.Smarandache n, Smarandache Sn m n m!. Sn = min{m : m N, n m!}. 96

104 Smarandache F.Smarandache Only Problems, Not Solutions,! Sn n = p α pα pα r r n, Sn = max }. Sn : i r {Spα i i S =, S =, S3 = 3, S4 = 4, S5 = 5, S6 = 3, S7 = 7, S8 = 4, S9 = 6, S0 = 5, S =, S = 4, S3 = 3, S4 = 7, S5 = 5, S6 = 6,. Sn,,!, Lu Yaming Sn,. k, Sm + m + + m k = Sm + Sm + + Sm k m, m,, m k. [5] Sn! Sn P n = ζ 3 3 x 3 ln x x 3 + O ln x P n n, ζs Riemann zeta-. [49] Smarandache, n n, d n Sd. PSn [, n] Sn n PSn., J.Castillo lim n n.,., Smarandache Sn,, F.Smarandache Sn!,,,.,! : 97

105 Smarandache 6.9 n >, PSn = + O n ln n J.Castillo. 6.5 n, PSn lim n n =. n PSn. n >, n = p α pα pα r r n, Sn Sn = S p α i i = m p i. α i =, m = Sn = p i. α i >, m >, Sn. n PSn [, n] Sn = Sn n! Sn = n =. M = ln n, n PSn = k n Sk=Sp α, α Sk M 6-39, + kp α n αp>m, α M <p n n p + n ln n + p n α> M kp n p>m p α n αp>m, α 3 n p α + kp α n αp>m, α 3 n p α p n p> M, α 3 n ln n + n + n M M n ln n n ln n + n p α M <p n p n αp>m, α p kp α n αp>m, α k n p + n p α + p α n αp>m, α 3 p n αp>m, 3 α<p 6-39, [ ]. M p M, αp = p, αp M p. 98 k n p α n p α 6-40

106 u = p M Smarandache p αp. Sk M k, Sk = Sp α, Sk p α M!, α j= [ ] M p j M p. Sk M k u k u, du. = + αp = [ ] M + p Sk M d u p M p M = exp [ ] M ln p p M expy = e y. [67] [] πm = p M = M M ln M + O ln, M p M M ln p = M + O ln M : p M = p M [ ] M ln + p p M [ ln p + M ln p ln ln + M p p ] πm lnm p M = M lnm ln M p ln p + p M M M + O = O ln M M ln M 6-4 M = ln n, : Sk M exp c ln n ln ln n 6-43 c. 99

107 Smarandache exp c ln n ln ln n n ln n, 6-39, : n PSn = + k n Sk=Sp α, α n = O ln n. n PSn = n + O ln n 6.0 n, fn fn : f = fn =, n > n = p α pα pα k k n : fn = maxα, α,, α k fn = minα, α,, α k. f =, f3 =, f4 =, f5 =, f6 =, f7 =, f8 = 3, f9 =, f0 =, ; f =, f =, f3 =, f4 =, f5 =, f6 =, f7 =, f8 = 3, f9 =, f0 =,. Smarandache m, n, m, n =, fmn = max{fn, fm} fmn = min{fm, fn},, x >, fn = bx + Ox +ɛ k p b = ζk + p k+ k= p ζs Riemann zeta-. 00, ɛ,

108 Smarandache 6. x >, ζ 3 fn = ζ3 x + O x 3 ln x p x >, An = {, n kk ; 0,. n,p= An = ɛ. fs = n= n,p= p k p ζk + p k+ x + Ox +ɛ An n s, Euler An fs = Aq m q ms = + q s + q s + q ks q p m=0 q p = q k+s q p q s = = ζs ζk + s + p s + p s + p ζs ζk + s pks p s p k+s ks ζs Riemann zeta-. Perron, s 0 = 0, T = x, b = 3 An = 3 +it p ks p s x s πi p k+s s ds + O x 3 T n,p= ζs 3 it ζk + s 0

109 Smarandache 3 ± it ± it, πi 3 +it ζs 3 it ζk + s fs = p ks p s x s p k+s s ds ζs p ks p s x s ζk + s p k+s s s = ζk+p k+. 3 +it +it it 3 it ζs p ks p s x s πi 3 it 3 +it +it it ζk + s p k+s s ds p k p x = ζk + p k+ +it πi + 3 +it n,p= it it An = 3 it it pk p x ζs p ks p s x s ζk + s p k+s s ds x +ɛ p k p ζk + p k+ x + Ox +ɛ 6.0. B Square-full, n B = ζ 3 ζ3 x ζ + 3 ζ x 3 + O x 6 ζs Riemann zeta-. [0] C Cubic-full, = Nx 3 + O x 4 n C 0

110 Smarandache N. [5] n >, n = p α pα pα r r, A, B, C k., fn = fn + fn n A n B n A, fn fn = = 0, n A n A n B, fn = k, p k, 6.0. fn = n B fn=k, n B = p k x k = = x = x = x k= k n x p k n, p=, n C p x k k k [ p ζk + p k+ x + O k= k= k= k ζk + p x k k ζk + p p p k+ + O x +ɛ p p k+ p>x k k p ζk + p k+ + O x +ɛ p x +ɛ] p k p p k+ + O x +ɛ 03

111 b = k, k= Smarandache k= k ζk + p k p ζk + p k+, p fn = bx + O x +ɛ p p k+, ɛ, ζs Riemann zeta B Square-full, C B, D Cubic-full n, fn = 0, fn = n B fn + fn n C fn + fn = = n B\D n D + O n B = n B 6.. n D + O n D n D ln x ln = ζ 3 ζ3 x + O x 3 ln x ln x ln 04

112 Smarandache Smarandache 7. Smarandache,,,.,. 7. Smarandache ax by + c = 0 : ax by + c = 0, a, b N c Z. Pell s x Dy =., : a b, ; x n, y n.. [ Gaceta Matematica, Volumen, Numero, 988, pp.5-7. Francisco Bellot Rasado :.Un metodo de resolucion de la ecuacion diofantica..] 7.. ax by + c = 0 ab = k k N, : ax kyax + ky = ac. a, b c,.. x 0, y 0, x, y 0 x 0 < x. 3: { x n+ = αx n + βy n ; y n+ = γx n + δy n, 05

113 Smarandache 3. : aαβ = bγδ 4; aα bγ = a 5; aβ bδ = b 6, α, β, γ, δ. 5 6 α, β 4 α, β, 7: aδ bγ = a : α = ±δ 8., β = ± b a γ 9., 5 8, 6 9 : aα bγ = a 0., x n+ = α 0 x n + b a γ 0y n ; y n+ = γ 0 x n + α 0 y n, α 0, γ 0 0 αγ 0. A = α 0 b a γ 0 γ 0 α 0, x, y, x A y 06

114 Smarandache A x y, A A, AA = A A = I.,,. x n, y n = x n, y n. GS : = A n x n x 0 y n y 0 A 0 = I, A k = A A k A. GS = A n x n x 0 y n y 0 GS x n = A n x 0 y n y 0 reduction ad absurdum, GS. u, v. k 0 N: u, v = A k0 x 0 y 0 k N : u, v = A k x 0 y 0 u, v GS., u i+, v i+ = A i =,,, u 0 = u, v 0 = v. i, u i+ < u i, x 0 < u i0 < x, 0 < u i0 < x 0,. u i v i 07

115 Smarandache GS 3 x n y n = A n x 0 εy o n N, ε = ± GS3. λ. Det A λi = 0 λ, V, : Av i = λ i v i, i,. t P = A n v M R v P AP = λ 0 0 λ. A n = P λ n 0 0 λ n GS3 GS3.. x 3y = 5 x n y n = n 3 λ, = 5 ± 6, v, = 6, ±, n N, x n y n : x n = 4 + ɛ n + 4 ɛ n ; y n = 3ɛ n + 3ɛ n. 6. x 3y = 0, : x n = + 3 n + 3 n ; y n = + 3 n + 3 n. 3

116 Smarandache n,, 0, 4,, 4, 8, 5, 30. : 3. x y + 3 = 0. : n x n 7 4 = 7 y n 3 ɛ =? 4. x 6y 0 = 0. : x n y n 5. x y 9 = 0. : x n y n = = n n 4 ɛ 3 0 =? =? fx, y = 0, : ax + by + c = 0. a, b, c Z. ab 0. : x + 6xy 3y 6x 6y + 0 = 0 4 u 7v + 45 = 0, 4. : 5 u = 3x + y v = y +. 09

117 Smarandache { u n+ = 5u n + 8v n ; v n+ = 8u n + 5v n. 6 u 0, u 0 = 3, 3ɛ. : : n N v n, u n v n 3 n +, : v n+ = 8u n + 5v n = even + odd = odd, u n+, v n+ 3 u n, v n 3., x n, y n, n N, 5 : x n = u n v n + 3 ; 6 y n = v n. 7 GS3 4, 7 3. : 3 GS3, 5 : u n = 3x n + y n, v n = y n +., 6, 8: x n+ = x n y n + 3 ; y n+ = x n + 9y n + 3. x 0, y 0 =, or,.. A = n N. 0 x n y n = A n 9

118 Smarandache 8 y n+ y n y 0 mod3, x n Z., : n a i x i = b 0, i= a i, b Z. a i, a j 0, i j < n, 0., i 0, j 0,, n a i0 a j0 < 0., n N. : x n+ h = n i= a ih x n i, h n x 0,, x 0 n 0. 0, n, a ih, i, h n.,., - A n. ax +by cz +d = 0, a, b, c N, d Z. 7.3 N Smarandache N,. V. Popa s [80], I.Cucurezeanu s [78] 65, Clement s S.Patrizio s [79].,,. 7. A, B. AB 0modpB A 0modp A p.

119 Smarandache 7. p, q, a, p, b, q. A 0modp, B 0modq aaq + bbp 0modpq aa + bbp q aa p + bb q 0modp 0modp., A = K p, B = K q, K, K Z, aaq + bbp = ak + bk pq. aaq + bbp = Kpq, K Z, aaq 0modp, bbp 0modq, A 0modp, B 0modq p,, p n a,, a n i, a i, p i. A 0modp,, A n 0modp n P D n i= a i p i p i 0mod P D n a i A i p j 0modp p n i= j i P = p p n D p, n i= a i p i p i.

120 Smarandache 7.3 p ij, i n, j m i, r,, r n, a,, a n i, a i r i. i p i,, p in c i omodr i, i. p ij, i n, j m i R D n i= P = n r i D R. i= a i c i r i 0mod R D i, r i m i p ij,, P = n,m i i,j= P D n i= j= a i c i m i 0mod P D p ij j= p ij D P. n a i c i i= P m i 0modP p ij j=,. n i= a i c i m i p ij j=,, m + + m n. a,, a n r,, r n, D, c,, c n. 3

121 Smarandache C i, Wilson Leibniz Smarandache[8], p p k!k! k 0modp, p k, k = p+ m, x x, p k!k! ],, C i,., C i, C i,, C h,.,. m i =, C i Simionov s i. a D =, V.Popa s, Cucurezeanu s, Clement s. b D = P P, a i, k i, i =,, 3, S. Patrizio s. p, p,, p n, i, k i p i. p, p,, p n T U V n [p i k i!k i! k i ] p i 0 mod p p p n i= j i n [p i k i!k i! k i ] i= p i j i p s+ p n 0 mod p p p n n [p i k i!k i! k i ] p j i= p i 0modp j 4 W n [p i k i!k i! k i ] i= p i

122 Smarandache. [8] : p p 3! p 0modp [ n pi 3! p ] i p 0modp p i i= p p + p!3p + + p + 0modpp + p!p 0modpp + p! + p + p! + p +. Clement s p!4+p+4 0 mod pp+. p, p + k, : p p + k I.Cucurezeanu s [78], P 65 : k k![p! + ] + [k! k ]p 0modpp + k p, p +, p + 6, p + 8: p! + p. + p!! + p + + p!6! + p + 6 p, p, p + 4, : + p!8! + p + 8 p! + p [p 3! + ] p + p [p + 3! + ] p + 4 modp 5

123 Smarandache S.Patrizio s 8p + 3! p p 3! p modp. 7.4 Smarandache Smarandache : k, y = x x x k +, y x i, i =,,, k.,.,.,, p m = p p p n +, p =, p < p < < p n, m > n. : + = 3, 3 + = 7, = 3, =, = 3,,..84: n, p i, i =,,, n, m > n. p m = p p p n + 7-6

124 Smarandache..85: m = n, m = n, m = nn +,.86: y = x x x k + 7-, k x x x k. k,. 7.5 Andrica Andrica Smarandache. p x n+ p x n =, 0.5. p n n , : n =, x =. : 3 x x =. n = 3, x = = a 0. : 7 x 3 x =. Andrica A n = p n+ p n < : B n = p a n+ p a n < 3. a < a 0. Andrica, x 7 x = x, a 0 = Andrica, 3 B n. : ;,, x : 3 x x =, x = x 3 x =, x x 5 x =, x x 7 x =, x x x =, x

125 Smarandache 7 x 3 x =, x x 7 x =, x x 3 x =, x x 3 x =, x x 89 x =, x x 3 x =, x x 39 x =, x x 8 x =, x x x =, x x 93 x =, x x 37 x =, x x 467 x =, x x 509 x =, x x 53 x =, x x 743 x =, x x 773 x =, x x 839 x =, x x 863 x =, x x 887 x =, x x 953 x =, x x 99 x =, x x > a 0, x. :

126 Smarandache C n = p k n+ p k n < k, p n n, k D n = p a n+ p a n < n 4. a < a 0 n, n = na. a a <? b a n n 0 n n 0 4? : p n+ p n 5/3 5, n =., p n+ p n. 6, p n p n+ /6, n =. 7.6 Christianto-Smarandache, , Google, 9

127 Smarandache. Poly-emporium Theory :Poly, emporium,.,.,., A.Cournot Cournot,,.. 883, Joseph-Bertrand Bertrand,,,.. Bertrand.,..,.,,. Thomas More , A.Cournot..,,,.,....,,. 0

128 Smarandache, n, F,F,...,F n,n 3,?,,. 3, :,..,,,.,,.,,.,,,,, /.. 3,., n..?,.,,,.,.,, F, F,...,F n,.,,...,,,..,,., William Petty Francois Perroux ,.

129 Smarandache.,.,,.,,.,,,,,,,.,,.,.,.,., ,.., J.M.Keynes98 936, 0 60, 70 James Tobin 98, :,,. 8,,... Robert Torrens David Ricardo77-83,,,,,..

130 Smarandache,,,,,.. n,.,,,,,,.,,,.,,.,,,., n,..,,,,,,,.,, ;, n,,,., Smarandache 7.7. L,. Wilson, Fermat, Euler, Gauss, Lagrange, Leibniz, Moser Sierpinski p, A = m Z m = +p β, p β, β N, m = + α, α = 0,,, m = 0. m = εp α...p αr r, ε = +, α i N, p,..., p r. 3

131 Smarandache L : Z Z, Lx, m = x + c x + c φm. c,..., c φm m, φ Euler. p i x m. : m A, m / A, Lx, m ±mod p i α i...p i αr i r. α Lx, m ±0 mod m/p i i...p i αr i r. d = p i α i...p i αr i r m = m d Lx, m ± + k 0 d k 0 m modm, k 0, k 0 k m k d = ± A m..c c φm ±modm, m A, m A. Wilson p, p! modm. :. c,, c φpα p α, p α N, k Z, β N, kp β + c,, kp β + c φpα p α p α., i φp α kp β + c i p α. p α i i. c,, c φpα m m, p α i i m, c,, c φpα p α i i φ m α i p i. m 3. c,, c φq b q, b + c,, b + c φq q Ô. 4

132 Smarandache, b, q b = c i0 = q b, b + c i = Mq q. :. x, m/p i α i pi r α ir =, x + c x + c nm 0mod m/p i α i p i r α ir. 4. c c φm ±modm, i, m A m / A c c φm ±modp α i i. 5. p i x m, m A m / A c c φm ±modp α i i.,, 4, x + c x + c φm c c φm ±modp α i i. 5 :. p i,, p i r, x, m, x + c x + c φm ±modp α i i. Lx, m = ± + k d = k m, k, k Z. d, m =, k m k d = ± k, k. k = m t + k 0 k 0 = dt + k 0, t Z, k 0, k 0. : Lx, m ± + m dt + k 0 modm Lx, m k 0 modm Lagrange Wilson : p, x p x + x + x + p modp; 5

133 Smarandache : m 0, ±4, x + s 0, x φms+s x s x + x + x+ m modm, m s s : { { x = x 0 d 0 ; x 0, m 0 = m = m 0 d 0 ; d 0 d 0 = d 0d ; d 0, m = m 0 = m d ; d { d s = d s d s ; d s, m s = m s = m s d s ; d s 7-5 { d s = d s d s ; d s, m s = m s = m s d s ; d s 7-6 [88], [89]. m, m s = m, s = 0 φm = m, Lagrange. L.Moser : p, p!a p + a = Mp, Sierpinski [87], P.57 : p, a p + p!a = Mp, Wilson, Fermat. L, : a m, m 0 c,, c φm m, c c φm a φm s+s L0, ma s = Mm L0, ma φms+s + c c φm a s = Mm,, x + c x + c φm a φms+s Lx, ma s = Mm 6

134 Smarandache Lx, ma φm s+s + x + c x + c φm a s = Mm, Fermat, Euler, Wilson, Lagrange Moser. Moser Sierpinski [9], 7.40, PP.73-74, : m, m 0, 4, a, a m am! = Mm, Fermat Wilson. Leibniz : p, p! modp ; c i i+ c, c i i+ c, 0 c i < m, 0 c i+ < m c i c i modm, c i+ c i+ modm; c, c,, c φm m i, m 0, c < c i+ modm, m A m / A c c c φm ±modm, c φm modm. 7.8 Smarandache G,,,,? 8, G,,, / / / /,.. G I G = {g, g,, gk, }, G : SG = {a i : a = g, a k = a k 0 + g k }, k H. Ibstedt SMARANDACHE University of Craiova, Romania,August -4, a G II : a. G = {, 3, 5, 7, 9,, } : 7

135 Smarandache, 3, 35, 357, 3579, 3579, 3573,. 00 5,, 5, 7, 63, 93.? a. G = {, 4, 6, 8, 0,,...} :, 4, 46, 468, 4680, 4680, 00, n,, p, p,. : n! a.3 G =, 3, 5, 7,, 3, 7, :, 3, 35, 357, 357, 3573, 35737, 4 ; , 00. :? H. Ibstedt I t : {a i }, a i a i > a i t. QBASIC, t = H. Ibstedt. 3 s, s, s 3,..., s n, S. s, s s, s s s 3,. H. Ibstedt,, S? 8

136 Smarandache 4 f, R k. k n, n,, n k n + n + + n k = n, n : n = Rfn, fn,, fn k, : m? n, 5 / / / / /. : / / / / /,.,,.,? :, S, S.,? 6 SPDS,. : 506 = 56036, 56/0/36. C. Ashbacher SPDS., SPDS. : 44 SSPDS, 44 = 9448 SSPDS. m, m, m 4 SSPDS? : SSDPS, : = 44 3 = 69. SSDPS?? 7 n a I 9

137 Smarandache N, N N. N = absn N. : N3 = absn N,. N, N 0 : 5, 05, 50.,., : N, 0, 0, 0, 5,, 9, 8, 63, 7, 45. b n Smarandache II N n, N n. N = absn N. : N3 = absn N,. N n, N 0 k, where N N, k. : 4 5, 0004, Dirichlet, Smarandache 3 00 N 999: - 90, 0,,,, N = 0; -, : 99, 89, 693, 97, 495. Smarandache N 9999: - 8. N = 09 ; : 78, 6534; or 90, 80, 630, 70, 450; 909, 88, 6363, 77, 4545; or 999, 899, 6993, 997, 4995; - 0. H. Ibstedt 8 Erdos-Smarandache : P n = Sn :, 3, 5, 6, 7, 0,, 3, 4, 5, 7, 9, 0,,, 3, 6, 8, 9, 30, 3, 33, 34, 35,. P n n, Sn Smarandache : Sn! n. 30

138 Smarandache 7.9 Smarandache Ceva, Ceva, Ceva : ABC AA, BB, CC, A B A C BC B A CA C B = A A A n, M, p = n n 3 n i j, j {i + s,, i + s + t }, M ij A i M A i+s A i+s+,, A i+s+t A i+s+t., M ij A n, s + t = n, s, t 0. n, i+s+t i, j=, i+s M ij A j M ij A pj = n, M ABC,., M, A i. Ma, b A i X i, Y i, a, b, X i, Y i i {,,, n}. : i {,,, n}, X i a 0, Y i b 0. A i M i n : x a X i a y b Y i b = 0., dx, y; X i, Y i = 0. M ij A j = δa j, A i M M ij A pj δa pj, A i M = dx j, Y j ; X i, Y i dx pj, Y pj ; X i, Y i =, Dj, i Dpj, i, 3

139 Smarandache, δa, ST A ST, Da, b dx a, Y a ; X b, Y b., : a + b pp pa, }{{} b a b p p p a. }{{} b i+s+t j=i+s M ij A j M ij A j+ = = = i+s+t j=i+s Dj, i Dj +, i Di + s, i Di + s +, i Di + s, i Di + s + t, i Di + s +, i Di + s + t, i Di + s +, i Di + s + t, i = Di + s, i Di s, i. n Di + s, i Di s, i 3 i= +, s + Ds D, s + + t +, s + t + Ds Dt +, s + t + D + s, D + s, = D, + s n = i= Di + s, i n Di, i + s = Dr, p Dp, r = D + s, D + s, Ds, s Ds +, s + = D s, D s, Dn, s D, s + Ds + t, s + t Ds + t +, s + t + Dt, s + t Dt +, s + t + Ds + t + s, s + t + s Dt + s, s + t + s Ds + t, s + t D, + s i= X r a X p a Y r b Y p b X p a X r a Y p b Y r b P i + s P i Ds + t, s + t = n. = X r ay r b X p ay p b Ds, n Dn, s = P r P p

140 Smarandache, X t ay t b = P t. t {,,, n}, P t t A i0 M ; A i A i+, a i, s + A M a s+ A M. : s = 5, t = 3, : - A M A 6 A 7, A 7 A 8, A 8 A 9. - A M A 7 A 8, A 8 A 9, A 9 A 0. - A 3 M A 8 A 9, A 9 A 0, A 0 A. : M ija j M ij A pj M A A A k+, i {,,, k + }, M i A i A pi M n M i A i. M i {A i, A pi }, =. M i= i A pi s = k, t =, n = k +,... : 5. A M 3, A M 4, A 3 M 5 M. K = M 3A 3 M4A 4 M5A 5 M 3 A 4 M 4 A 5 M 5 A A 4 M M, : M A M A = K K,,A 4M M. 7-7 A 5 M, M A = M A 3. : 5 M i A i =. M i A pi i= 33

141 Smarandache , i j, j {i, p i}, M ij = A i M A j A pj, M ij n M ij A j {A j, A pj }, = n. M ij A pj i, j= s =, t = n, s + t = n Ceva. n = 3, s =, t =, Ceva 7.9. A A A n, s, t 0 s + t = n. A i d i, A i+s A i+s+,, A i+s+t A i+s+t M i,i+s,, M i,i+s+t, n M i, i+s+t M ij A j n M i = A i+s M ij A j+ M i A. i+s+t i= j=i+s i, M i,i+s. A i M i,i+s A i+s M i M i,i+sa i+s+, M i,i+s A i A i+s M i,i+s A i+s+ M i, A im i,i+s A i+s A i+s+ M i,i+s M i. M i,i+s A i M i,i+s A i+s+ = i= A ia i+s. 7-8 M i A i+s+ 34

142 Smarandache, M i,i+s A i A i+s+ M i,i+s A i+s M i, M i,i+s A i = A ia i+s+ M i,i+s A i+s M i A. 7-9 i+s M i,i+s A i+s M i,i+s A i+s+ = M i A i+s M i A i+s+ A i A i+s A i A i+s M i,i+s,,, 3. : i+s+t j=i+s M ij A j M ij A j+ = i+s+t j=i+s M i A j A i A j M i A j+ A i A j+ = M i A i+s M i A M i A i+s+ i+s+ M i A M i A i+s+t A i A i+s i+s+ M i A AiA i+s+ AiA i+s+t i+s+t A i A i+s+ A i A i+s+ A i A i+s+t = M i A i+s A i A i+s M i A. i+s+t A i A i+s+t n M i, A i+s A i A i+s i= M i A = i+s+t A i A i+s+t n M i A i+s, M i A i+s+t i= n A i A i+s = A A +s A A +s A s A s A i A i+s+t A A +s+t A A +s+t A s+ A i= 35

143 Smarandache As+A s+ A s+ A s + t = n. As+tA n A s+t A t A s+t+ A A s+t+ A t+ A s+t+ A A s+t++ A t+ A n A s A n A s+t =, A A A s, A i d i, A i+s A i+s M i, M i,. n i= M i A i+s M i A i+s = n M i A i+s M i A. i+s i= t =, n s = n +. s =, [] , d i, n i= M i A i+s M i A i+s+t = n. 7.0 Smarandache, Einstein-Podolski-Rosen Bell, :,,? :,,?. :? Innsbruck 997 Rainer Blatt, David Wineland et al. - ; - ; -,., ; 36

144 Smarandache -,,, ; - A B, A, B A B A B,, Einstein-Podolski-Rosen Bell ; - Nicolas Gisin., 55 ; - Rupert Ursin 600 ;..,, A, A B. 3.? Smarandache,,,,,,. Kamla John Scott Owens 00 Hans Gunter Smarandache,., c. c c c c? 37

145 Smarandache. v w : v < c w = c v = c w = c v > c w = c v > c w > c v < c w = v = c w = v > c w = v = w =? 7. Smarandache, QCD, : Q A ±M3, 7- M3 3., ±M3 = {3 k k Z = {,, 9, 6, 3, 0, 3, 6, 9,, }, Q, A 5 Q Amod 3 7- Q A 3., 3, 3 M3, 6, 9,., 3, 3 M3, 6, 9,.,,, 3,,, 5, 3,,,

146 Smarandache q, {,,,,, }, a, {,,,,, }., n, n,, : n =, qa - n = 3, qqq, aaa - n = 4, qqaa n = 5, qqqqa, aaaaq n = 6, qqqaaa, qqqqqq, aaaaaa n = 7, qqqqqaa, qqaaaaa n = 8, qqqqaaaa, qqqqqqaa, qqaaaaaa n = 9, qqqqqqqqq, qqqqqqaaa, qqqaaaaaa, aaaaaaaaa n = 0, qqqqqaaaaa, qqqqqqqqaa, qqaaaaaaaa. 7. Smarandache..,. N V A. N V A., A A,., N, N N, N. V V,A, A,. NV,NV., :. N N. A A. V V

147 Smarandache N N. V V. A A =..,.. 3. A A ? 4. N A A.. 40

148 Smarandache ,.,,. 5. N V, V. :,.,.,.,.,.,., V, V. :,.., ,. 7.N N. :.. 4

149 Smarandache V V. : N N. :.. :...,..,. [ ]. 0. a N N. :

150 Smarandache. [, ]. 0. b N N. :..., [ ].. A A. :..... /...[ ].. V V. :.... [ ].... [ ]. 43

151 Smarandache. 3. V V. :. :[ ]. [ ] N N. : A A. : [ ]. [ ]... [ ]., N N. :.[ : ]..... [ ].. 44

152 Smarandache [ ]. [ ]. 7. A A. : V V. :....,.... [ ]. : :,!.,,.,. 7.3 Smarandache Smarandache.. 45

153 Smarandache.,,.,,,. Russell,Cantor, Grelling,,Skolem,.[Smarandache : A, A,A, A.,. Weisstein,998].,. Eubulides of Miletus.,,,.,A A.A A?. :,,... : /, /, /, /,,... :... :. a,,.,,.. b.,., 46

154 Smarandache,,.... :,, Heisenberg... :,..3. :...3. :,..4. :.4. :,. 3, Smarandache. M.Khoshnevisan,,,,. 47

155 Smarandache [] F. Smarandache, Only Problems, Not Solutions [M], Chicago, Xiquan Publishing House, 993. [] Farris Mark and Mitchell Patrick, Bounding the Smarandache function [J], Smarandache Notions Journal, Vol. 300, [3] Liu Yaming, On the solutions of an equation involving the Smarandache function [J], Scientia Magna, Vol. 006, No., [4] Jozsef Sandor, On certain inequalities involving the Smarandache function [J]. Scientia Magna, Vol. 006, No. 3, [5], Smarandache [J],, Vol.49006, No.5, [6] Le Maohua, An equation concerning the Smarandache LCM function [J], Smarandache Notions Journal, 004, 4: [7] Lv Zhongtian, On the F.Smarandache LCM function and its mean value [J], Scientia Magna, 007, 3: -5. [8], F.Smarandache LCM [J],, 007, 33, [9], [J],, 007, 3, [0],. [M],,, 003. [], [M],,, 007. [] Wang Yongxing, On the Smarandache function. Research on Smarandache Problem in Number Theory Edited by Zhang Wenpeng, Li Junzhuang and Liu Duansen. Hexis, Vol.II005, [3] Tom.M. Apostol, Introduction to analytic number theory, New York: Springer- Verlag, 976, 77. [4] Kenichiro Kashihara, Comments and topics on Smarandache notions and problems, Erhus University Press, USA, 996. [5] Guohui Chen, Some exact calculating formulas for the Smarandache function, Scientia Magna, Vol.006, No., [6] C.Ashbacher, Some properties of the Smarandache-Kurepa and Smarandache- Wagstaff functions, Mathematics and Informatics Quarterly, 7997, 4-6. [7] A.Begay, Smarandache ceil functions, Bulletin of Pure and Applied Sciences, 6E997, 7-9. [8] I.Balacenoiu and V.Seleacu, History of the Smarandache function, Smarandache Notions Journal, Vol , 9-0. [9] A.Murthy, Some notions on least common multiples, Smarandache Notions Journal, Vol. 00, [0] Aledsandar Ivić, The Riemann zeta-function theory, New York, 985, [] Fu Jing, An equation involving the Smarandache function. Scientia Magna, Vol. 48

156 006, No. 4, [] F. Smarandache, Sequences of numbers involving in unsolved problem, Hexis, 006, 7-8. [3] M.L.Perez, Florentin Smarandache, Definitions, solved and unsolved problems, conjectures and theorems in number theory and geometry, Xiquan Publishing House, 000. [4] Jin Zhang, Pei Zhang, The mean value of a new arithmetical function, Scientia Magna, Vol.4, No., 008: [5] Aleksandar Ivić, The Riemann Zeta-function, Dover Publications, New York, 003. [6] Jozsef Sandor, On certain generalizations of the Smarandache function, Smarandache Notions Journal, Vol. 000, No.--3, 0-. [7] M.N.Huxley, The distribution of prime numbers, Oxford University Press, Oxford, 97. [8] A.Ivic. On a problem of Erdös involving the largest prime factor of n. Monatsh. Math., Vol.45005, [9] S. Tabirca. About S-multiplicative functions, Octogon, 999,7: [30] P.Erdös. Problem 6674, Amer. Math. Monthly, Vol.98, 99, 965. [3] Jozsef Sandor, On a dual of the Pseudo-Smarandache function [J], Smarandache Notions Book series, Vol.300, 6-3. [3] Maohua Le, Two function equations [J], Smarandache Notions Journal, Vol.4004, [33] M. Bencze, Aplica.ii ale unor.iruri de recuren...n teoria ecua.iilor diofantiene, Gamma Bra.ov, XXI-XXII, Anul VII, Nr.4-5, 985, pp.5-8. [34] Z. I. Borevich - I. R. Shafarevich, Teora numerelor, EDP, Bucharest, 985. [35] A. Kenstam, Contributions to the Theory of the Diophantine Equations Ax n By n = C. G. [36] H. Hardy and E.M. Wright, Introduction to the theory of numbers, Fifth edition, Clarendon Press, Oxford, 984. [37] N. Iv..hescu, Rezolvarea ecua.iilor.n numere.ntregi, Lucrare pentru ob!inerea titlului de profesor gradul coordonator G. Vraciu, Univ. din Craiova, 985. [38] E. Landau, Elementary Number Theory, Celsea, 955. [39] Calvin T. Long, Elementary Introduction to Number Theory, D. C. Heath, Boston, 965. [40] L. J. Mordell, Diophantine equations, London, Academic Press, 969. [4] C. Stanley Ogibvy, John T. Anderson, Excursions in number theory, Oxford University Press, New York, 966. [4] W.Sierpinski, Oeuvres choisies, Tome I. Warszawa, [43] F. Smarandache, Sur la r solution d quations du second degr a deux inconnues dans Z, in the book G n ralizations et g n ralit s, Ed. Nouvelle, F s, Marocco; MR:85h: [44] David Gorski, The pseudo Smarandache function [J]. Smarandache Notions Journal, 00, 3: [45] Liu Yanni, On the Smarandache Pseudo Number Sequence [J]. Chinese Quarterly Journal of Mathematics. 006, 04:

157 Smarandache [46] Lou Yuanbing, On the pseudo Smarandache function [J]. Scientia Magna, 007, 34: [47] Zheng Yanni, On the Pseudo Smarandache function and its two conjectures [J], Scientia Magna. 007, 34: [48],. [M]. :, 98. [49], Smarandache Sn [J],, 007, 3: [50], [J],, 007, 33: [5], Smarandache [J],, Vol.8 008, No., [5], Smarandache [J],, Vol.0 008, No.4, [53] Jozsef Sandor, On additive analogues of certain arithmetic function, Smarandache Notions J, Vol.4004, 8-3. [54] Sloane, N.J.A. Sequence A003/M096 in An On-Line Version of the Encyclopedia of Integer Sequences. [55] C.Dumitrescu and V.Seleacu, Some solutions and questions in number theory [M], Erthus Univ Press, Glendale, 994. [56] G.H.Hardy, E.M.Wright, An introduction to the theory of numbers [M], Oxford Univ. Press, Oxford, 98. [57] H.L.Montgomery and R.C.Vaughan, The Distribution of Squarefree Numbers, Recent Progress in Analytic Number Theory [M], Vol., [58] V.Mladen and T.Krassimir, Remarks on some of the Smarandache s problem, Part [J], Scientia Magna, Vol. 005, No., -6. [59] Xigeng Chen, Two problems about -power free numbers, Scientia Magna [J], Vol. 006, No., [60] Felice Russo, A set of new Smarandache functions,sequences and conjectures in number, 000. [6] Maohua Le, The function equation Sn = Zn, Scientia Magna, 005, No., [6] Ashbacher,Charles, On numbers that are Pseudo-Smarandache and Smarandache perfect, Smarandache Notions Journal, 4 004, PP [63] Maohua Le, On the Pseudo-Smarandache-Squarefree function, Smarandache Notions Journal, 300, [64] BALACENOIUI, SELEACU V. History of the Smarandache function, Smarandache Notions Journal, 999, 0: [65] F.Mark, M.Patrick, Bounding the Smarandache function, Smarandache Journal, [66] Jian Ge, Mean value of F.Smarandache LCM function [J], Scientia Magna, Vol.3 007, No., 09-. [67],, [M],,, 988. [68] Le Maohua. A conjecture concerning the Smarandache dual functions[j]. Smarandache Notions Journal, 0044:

158 [69] Li Jie. On Smarandache dual functions[j]. Scientia Magna, 006: -3. [70] G.H.Hardy and E.M.Wright. An introduction to the theory of numbers, Oxford University Press, 98. [7] R.L.Duncan. A class of additive arithmetical functions. Amer.Math. Monthly, 6996, [7] D.R.Heath-Brown, The differences between consecutive primes, Journal of London Math. Soc., Vol. 8978, No., 7-3. [73] D.R.Heath-Brown, The differences between consecutive primes, Journal of London Math. Soc., Vol. 9979, No., [74] D.R.Heath-Brown and H.Iwaniec, TO the difference of consecutive primes, Invent. Math., Vol , [75] I.Balacenoiu and V.Seleacu, History of the Smarandache function, Smarandache Notions Journal, Vol , 9-0. [76] D.R.Heath-Brown, The differences between consecutive primes [J], Journal of London Mathematical Soc., 978, 8 : 7-3. [77] D.R.Heath-Brown, The differences between consecutive primes [J], Journal of London Mathematical Soc., 979, 9 : [78] Cucurezeanu, I., Probleme de aritmetica si teoria numerelor, Ed. Tehnica, Bucuresti, 966. [79] Patrizio, Serafino, Generalizzazione del teorema di Wilson alle terne prime, Enseignement Math., Vol., nr. 3-4, pp , 976. [80] Popa, Valeriu, Asupra unor generalizari ale teoremei lui Clement, Studiisi cercetari matematice, Vol. 4, Nr. 9, pp , 97. [8] Smarandache, Florentin, Criterii ca un numar natural sa fie prim, Gazeta Matematica, Nr., pp. 49-5; 98; see Mathematical Reviews USA: 83a: [8],, Smarandache [M], High American press American, 006. [83],,, Smarandache [M], High American press American, 008. [84],,, Smarandache [M], High American press American, 008. [85] V.Christianto and F.Smarandache, Cultural Advantage for Cities: An Alternative for Developing Countries,InfoLearnQuest, 008. [86] Lejeune-Dirichlet, Vorlesungen ber Zahlentheorie, 4te Auflage, Braunschweig, 894, 38. [87] Sierpinski, Waclaw, Cestimsi ce nustim despre numerele prime, Ed.Stiintifica, Bucharest, 966. [88] Smarandache, Florentin, O generalizare a teoremei lui Euler referitoare la congruente, Bulet. Univ. Brasov, Seria C, Vol. XXIII, pp. 7-, 98; see Mathematical Reviews: 84j: [89] Smarandache, Florentin, G n ralisations et G n ralit s, Ed. Nouvelle, F s, Morocco, pp. 9-3, 984. [90] Smarandache, Florentin, A function in the number theory, An. Univ. Timisoara, SeriaSt. Mat., Vol. XVIII, Fasc., pp , 980; see M.R.: 83c: [9] Smarandache, Florentin, Probl mes avec et sans... probl mes!, Somipress, F s, 5

159 Smarandache Morocco, 983; see M.R.: 84k: [9] Mudge, Mike, Smarandache Sequences and Related Open Problems, Personal Computer World, Numbers Count, February 997. [93] Mudge, Mike, Smarandache Sequences and Related Open Problems, Personal Computer World, Numbers Count, February 997. [94] Smarandache, Florentin, Collected Papers Vol. II, University of Kishinev, 997. [95] Erdos, P., Ashbacher C., Thoughts of Pal Erdos on Some Smarandache Notions, Smarandache Notions Journal, Vol. 8, No. --3, 997, 0-4. [96] Sloane, N. J. A., On-Line Encyclopedia of Integers, Sequence A

160 Research on Smarandache Unsolved Problems Jianghua Li Department of Mathematics Northwest University Xi an, Shaanxi, 707 P. R. China Yanchun Guo Department of Mathematics Xianyang Normal University Xianyang, Shaanxi, 7000 P. R. China High American Press 009

161