关于 Smarandache 理论 及其有关问题 王 妤 西北大学数学系 苏娟丽 杨凌职业技术学院 张瑾 西安师范学校 High American Press 2008

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2 关于 Smarandache 理论 及其有关问题 王 妤 西北大学数学系 苏娟丽 杨凌职业技术学院 张瑾 西安师范学校 High American Press 008

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 1346, Ann Arbor MI , USA Tel.: (Customer Service) Peer Reviewers: Wenpeng Zhang, Department of Mathmatics, Northwest University, Xi an,shaanxi, P.R. China. Wenguang Zhai, Department of Mathmatics, Shangdong Teachers University, Jinan, Shangdong, P.R. China. Guodong Liu, Department of Mathmatics, Huizhou University, Huizhou,Guangdong, P.R.China. Copyright 008 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. Smarandache., Comments and Topics On Smarandache Notions and Problems, Kenichiro Kashihara Smarandache,,.,.,, Smarandache, Kenichiro Kashihara, Smarandache, Smarandache,,,,.,! I

5 Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Pierced Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache S(n) Smarandache Smarandache Smarandache p Smarandache Smarandache Goldbach-Smarandache Vinogradov-Smarandache Smarandache-Vinogradov Smarandache-Logics Smarandache-Position Smarandache Smarandache Smarandache Smarandache Counter Smarandache C(n) Smarandache G(n) II

6 1.31 Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Smarandache Kenichiro Kashihara Smarandache Dirichlet Smarandache Dirichlet Smarandache Smarandache p Diophantine III

7 Smarandache Smarandache Goldbach Numberical Carpet Syllabic Puzzle Code Puzzle triangular number Smarandache-Kurepa Smarandache-Wagstaff n Smarandache Smarandache Near-To-Primordial IV

8 Smarandache Smarandache 1.1 n, y y = f(n).,,,. Florentin Smaradache Smarandache,,. 1. Smarandache 1.1: 1, 3, 456, 7891, 3456, 78913, , ,, 1,,..., 9, n n. 1.1: n : n : 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, : n : n n, n n(n + 1) (mod 9). 1.3: 1

9 Smarandache 1.3 Smarandache 1.: n 0, d s (n) n. : 0, 1,, 3, 4, 5, 6, 7, 8, 9, 1,, 3, 4, 5, 6, 7, 8, 9, 10,, 3, : n d s (n) : n = 10k, k N, d s (n), d s (n + 1),..., d s (n + 9) 1. [ n d s (n) = d s (10 + n 10 10]) [ n 10] n 10. [ n 10], 1.4 Smarandache 1.3: n 0, d p (n) n. : 0, 1,, 3, 4, 5, 6, 7, 8, 9, 0, 1,, 3, 4, 5, 6, 7, 8, 9, 0,, 4, 6, : n d p (n) : n = 10k, k N, d p (n) = 0, d p (n + 1),..., d s (n + 9)).. ([ n ]) ( [ n d p (n) = d p n 10, 10 10]) [ n 10] n 10.

10 Smarandache 1.5 Smarandache Pierced 1.4: n 1, c(n) = 101 (10 4n n n + 1). : 101, , , ,. 1.6: Smarandache : c(n) 101 :, 10 4n 1 = (10 4 1)(10 4n n ) = (10 4 1) c(n) n 1 = (10 n + 1)(10 n 1) = (10 n + 1)(10 n + 1)(10 n 1). : (a) n >, c(n) 101, (10n + 1) (10 n 1). n >, 10 4n 1 10 n n 1,. (b) n = 1 n =, c(1) 101 = 1 c() 101 = = , c(n) : n, c(n) 101 : [17] ( [4]), : 1.1: n c(n)

11 Smarandache : k-free : k. n > 1, n k-free, p p n, p k n. -free ; 3-free (mod 9). a b(mod m), a n b n (mod m) n, 10 4n 4 1(mod 9), 10 4n 8 1(mod 9), 10 4n 1(mod 9). 1 1(mod 9)., n = 9k c(n) 101 = 104n n n(mod 9). c(n) n n n 0(mod 9). 9 n, c(n) Smarandache 1.5: n 1, P d (n) n, P d (n) = n d(n), d(n) n. n = 1,, 3, 4,, P d (n) = 1,, 3, 8, 5, 36, 7, 64, 7, 100, 11, 178, 13, 196,.... 4

12 Smarandache 1.8: P d : p, P d (p) = p, P d (n). 1.9: p, P d p k (k = 1,, ) : p, n = p m, n p k p k, k = 1,,. 1.10: P d (n) = n n. : n, P d (n) > n, P d (n) = n n 1 p, p. 1.11: p p 4 P d p 5 P d : p. n q p, q P d, n. p. n = p k, n p, p, p 3,, p k, k(k + 1) p 1 k,. p, p m k(k + 1) P d k = m. m = 4, 5 k. 1.7 Smarandache 1.6: n 1, p d (n) n n., p d (n) = n d(n) 1. n = 1,, 3, 4,, p d (n) = 1, 1, 1,, 1, 6, 1, 8, 3, 10, 1, 144, 1, 14, 15, 64, : n p d (n) = 1 : p, p d (p) = p, n p d (n) = 1. 5

13 Smarandache 1.13: n, n = p 3 n = pq, p, q. : (i) n = p m (1-1), p d (p m ) = p d (n) = n (1-1) m 1 k=1 p k = p m(m 1) = p m, m(m 1) = m, m = 3. (ii) n = pq, p d (pq) = pq. (iii) n. n = p α 1 1 pα pα k k n. k α > 1 k >, p d (n) > p α 1 1 pα pα k k (pα 1 1 pα 1 ) > n. p d (n) > p α 1 1 pα pα k k (pα 1 1 pα ) > n. (1-1) n = p 3, pq, p, q. 1.7: p d (n) = n Smarandache. 1.8 Smarandache 1.8: n 1, n Smarandache SSC(n) nk k. : 6 1,, 3, 1, 5, 6, 7,, 1, 10, 11, 3, 13, 14, 15, 1, 17,, 19,.... Smarandache,.

14 Smarandache 1.14: Smarandache,. : k n, k = k 1 p r, p, r, k 1 n n, k. k = k 1 p r, r 3, k 1 p r n n, k., k 1. k n, k p nk, p p n, p k. 1.9 Smarandache 1.9: n 1, n Smarandache SCC(n) kn k. : 1, 4, 9,, 5, 36, 49, 1, 3, 100, 11, 18, 169, 196, 5,.... Smarandache,. 1.15: SCC. : k n, k = k 1 p s, s 3, k 1 p s 3, k. k n, k p 3 nk, p p n, p k Smarandache 1.10: (x + c 1 ) (x + c F(m) ) m, m =, 3, 4,, x N, c i, 1 i F (m), F., : m =, x + 1 x 1(mod ), m = 3, x + 3x + x 1(mod 3), 7

15 Smarandache m = 4, x + 3 x 1(mod 4), m = 5, x x x + 50x + 4 x 4 1(mod 5), m = 6, x + 6x + 5 x 1(mod 6). 1.16: : (x + c 1 ) (x + c F(m) ) x F (m) 1(mod m). : (c i, m) = 1, (m c i ) = 1, (m c i ) c 1 c F(m). (x, m) = 1, x F (m) 1 0(mod m). (1-) (x + c 1 ) (x + c F(m) ) 0(mod m). (1-3) (1-) (1-3) F (m)-1. c 1 c F (m), A 1 x F (m) 1 + A x F (m) + + A F (m) 1 x + A F (m) 0(mod m).. A 1 = A = = A F (m) 1 0(mod m), 1.11 Smarandache Smarandache, : 1.11: n 1, Smarandache P p (n) n. :,, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19,. 1.1: n, Smarandache p p (n) n. : 8,, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17,.

16 Smarandache n p p (n) P p (n). 1.17: n p p (n) = P p (n). : p, p p (n) = p = P p (n). 1.18: : 1.13: I n = {p p () + p p (3) + + p p (n)}/n. 1.14: S n = {P p () + P p (3) + + P p (n)}/n : lim n (S n I n ). S n 1.18.: lim n I n , [18], : 1.: n > 1, S ( ) n = 1 + O n 1 3. I n 1.1: 1. S n lim = 1. n I n : 1.1: x > 1, (p n+1 p n ) x ε, p n+1 x p n n, ε. : [1] [13]. 1.: x, P x x+x 3 9

17 Smarandache : x, P n P n x. 1.1 (P n P n 1 ) x ε P n P n 1 x 3. x x + x 3 p : x > 1, P p (n) = 1 ( ) x + O x 5 3 n x (1-4) n x p p (n) = 1 ( ) x + O x 5 3. (1-5) : (1-4), (1-5). P k k. P p (n) P r, P r+1 P r n P p (n) = P r. P p (n) = P n (P n+1 P n ) n x = 1 P n+1 x P n+1 x = 1 P (x) 1 ( P n+1 P n) 1 P n+1 x P n+1 x (P n+1 P n ) (P n+1 P n ), (1-6) P (x) P (x) x. 1. ( ) P (x) = x + O x 3. (1-7) (1-6), (1-7) 1.1 P p (n) = 1 ( ) ( x + O x O x 3 +ε) 18 = 1 ( ) x + O x n x

18 Smarandache (1-4). p p (n) = n x P n x = 1 P n x = 1 P (x) + 1 P n (P n P n 1 ) ( P n Pn 1) 1 + (P n P n 1 ) p n x (P n P n 1 ), P n x 1.1, 1., (1-5). : : n > 1, 1.3 I n S n I n = {p p () + p p (3) + + p p (n)}/n = 1 [ 1 ( ) ] n n + O n 5 3 = 1 ) (n n + O 3 (1-8) S n = {P p () + P p (3) + + P p (n)}/n = 1 [ 1 ( ) ] n n + O n 5 3 = 1 ) (n n + O 3. (1-9) (1-8) (1-9). S n I n = ( 1 n + O n 3 ( 1 n + O n 3 ) ( ) = 1 + O Smarandache,. : n 1 3 ). Smarandache : 11

19 Smarandache 1.15: n, c(n) C(n) n n, p p () p p (3) p p (n + 1) p c(n) = p (n + ) p p (n + 3) p p (n + 1) p p (n(n 1) + ) p p (n(n 1) + 3) p p (n + 1) C(n) = P p (1) P p () P p (n) P p (n + 1) P p (n + 3) P p (n) P p (n(n 1) + 1) P p (n(n 1) + ) P p (n ) : c(4) = C(4) = 1 c() = C() = = 1; c(3) = = 0; c(5) = = 4; C(3) = = 188; C(5) = = 14; = 4; , : = 96. = 144.

20 Smarandache 1.1: n 6, c(n) = 0 C(n) = 0. 1.: q, c(q) 0 C(q) 0., 1.1, ( ): n 6, c(n) = 0 C(n) = 0. 1.,,.! 1.1 Smarandache Smarandache, : 1.16: n, Smarandache SISP(n) n. : 0, 1, 1, 1, 4, 4, 4, 4, 4, 9, 9, 9, 9, 9, 9, 9,. 1.17: n, Smarandache SSSP(n) n. : 0, 1, 4, 4, 4, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16,. n, SISP(n) SSSP(n). 1.19: SISP(n) = SSSP(n). : n, SISP(n) = SSSP(n). n, SISP(n) < n < SSSP(n). 1.0: : 1.18: S n = {SSSP (1) + SSSP () + + SSSP (n)}/n. 13

21 Smarandache 1.19: I n = {SISP (1) + SISP () + + SISP (n)}/n : lim n (S n I n ). S n 1.0.: lim. n I n 1.1: : 1.0: s n = n SSSP (1) + SSSP () + SSSP (n). 1.1 i n = n SISP (1) + SISP () + SISP (n) : lim n (s n i n ). s n 1.1.: lim. n i n : , ( ): 14 (a) x >, n x n x ( ) SSSP (n) = x + O x 3 ; (1-10) ( ) SISP (n) = x + O x 3. (1-11) (1-10) (1-11)

22 Smarandache (b) n, (c) n, S ( ) n = 1 + O n 1 I n S n lim = 1. n I n s n i n = 1 + O ( ) n 1 s n lim = 1, lim n i (s n i n ) = 0. n n 1.0.1,,! 1.13 Smarandache 1.: n, SPAC(n) n+k k. : 1, 0, 0, 1, 0, 1, 0, 3,, 1, 0, 1, 0, 3,, 1, 0, 1, 0, 3,, 1, 0, 5, 4, 3,,. Smarandache k k, k 1, k, k 3,,, 1, 0. 1.: k k, k 1, k, k 3,,, 1, 0. : [19] : 1.3: k, SPAC(n). k, k 1, k, k 3,,, 1, 0 15

23 Smarandache : k, n > k +1. p p > n!+n. p 1, p,, p K,, n! + n,, n! +,. k + 1 : p k, p k + 1, p k +,, p 1, p. Smarandache SPAC(p k) = k, SPAC(p k 1) = k 1,, SPAC(p 1) = 1, SPAC(p) = 0. k, k 1, k,, 1, 0 SPAC(n),. 1.3: A n = {SPAC(1) + SPAC() + + SPAC(n)}/n. 1.3: lim n A n 1.3: {A n }. : [0], : 1.4: n, A n = 1 n n SPAC(a) 1 ln n + O (1). a=1, 1.4: n, n [n n 7 1, n] [n, n+ 1 ]. p q n 7 n n 7 1 p n n < q n + n

24 Smarandache : [1] [13]. 1.5 π (x) x, π (x) = x ( ) x ln x + O ln. x : [5] [7].. : n, = p 1 < p < p 3 < < p m n [1, n]. SPAC(a) (p i, p i+1 ] a p i <a p i+1 SPAC(a) = p i+1 p i 1 + p i+1 p i = (p i+1 p i )(p i+1 p i 1). SPAC(1) = 1, SPAC(a) = 1 + SPAC(a) + a n p i+1 n p i <a p i+1 (p i+1 p i )(p i+1 p i 1) p m = = p i+1 n = 1 = 1 p i+1 n p i+1 n p i+1 n p i+1 n (p i+1 p i ) 1 p i+1 n p m <a n (p i+1 p i ) SPAC(a) (p i+1 p i ) 1 (p m ). (1-1) (p i+1 p i ) (p i+1 p i ) 1 p i+1 n (π(n)) 1. (p i+1 p i ) 1 p i+1 n

25 Smarandache p i+1 n (1-1) A n (p i+1 p i ) (p m ). π(n) na n 1 (p m ) 1 π(n) (p m ) = 1 [ ] (p pm m ) 1 π(n) A n 1 [ ] n (p pm m ) 1. π(n) n p m n 7 1 A n = [ ( )] n + O n 7 1 n [ n ln n + O ( )] n + O ( n + O ( n ln n + O n 19 1 n ln n ln n ) = 1 ln n + O (1). ) + O(1) ( pm n ) [ ] 1 lim ln n + O (1) = +, n lim A n = +. n A n.. 1.4: n, n + k, k k. : 18 1, 0, 0, ±1, 0, ±1, 0, 1, ±, 1, 0, ±1, 0, 1, ±,

26 Smarandache. 1.4:. : ±k +k k, n Smarandache S(n) 1.5: n, S(n) n m! m. : 0,, 3, 4, 5, 3, 7, 4, 6, 5, 11, 4, 13, 7, 5, 6, 17, 6, 19, 5, 7,. Smarandache.,, Smarandache C.Ashbacher. 1.5: Dirichlet n=1 S(n) n s. 1.6: m k > 5 S(m) < S(m + 1) < < S(m + k), S(m) > S(m + 1) > > S(m + k). 1.7: 1 S(n 1) + S(n 1) + + (n 1) S(n 1) + 1 0(mod n). : [1], : 1.5: n, n 1 S(n 1) + S(n 1) + + (n 1) S(n 1) + 1 0(mod n) (1-13) 19

27 Smarandache n =, 3, 5. : (1) n =, (1 S(1) + 1), n = (1-13). () n = 3, 3 (1 S() + S() + 1) = 6, n = 3 (1-13). (3) n = 5, 5 (1 S(4) + S(4) +3 S(4) +4 S(4) +1) = 355, n = 5 (1-13). (4) n = p 7, p. g p,, 1 i p, (g i 1, p) = 1, m, g p 1 1(mod p) g m(p 1) 1(mod p) (1-14) p g 0, g 1,, g p p. 1 S(n 1) + S(n 1) + + (n 1) S(n 1) g 0 S(p 1) + g 1 S(p 1) + + g (p ) S(p 1) g(p 1) S(p 1) 1 (mod p). (1-15) g S(p 1) 1 p 7, S(p 1) p, (g S(p 1) 1, p) = 1. (1-14) (1-15) 1 S(n 1) + S(n 1) + + (n 1) S(n 1) g(p 1) S(p 1) 1 (mod p). g S(p 1) 1 1 S(n 1) + S(n 1) + + (n 1) S(n 1) + 1 1(mod n)., p 7, (1-13).. 1.8, 1 S(n) + S(n) + 3 S(n) + + (n 1) S(n) 0(mod n), ( ): 0

28 Smarandache (a) n > 1, µ(n) 0 1 S(n) + S(n) + 3 S(n) + + (n 1) S(n) n 4 (1 + ( 1)n ) (mod n), µ(n) Möbius. (b) n > 1 n = p α 1 1 pα pα k k, S(n)=S(pα j j )= αp j. n i = 1,,, k, φ(p α i i ) αp j. (a) (b), (a) : (c) n > 1 µ(n) 0, 1 S(n) + S(n) + 3 S(n) + + (n 1) S(n) 0(mod n). 1.8:. 1 S(a) + 1 S(b) = 1 S(c) (1-16) : [], : 1.6: (1-16). (a, b, c) S (a) + S (b) = z, S(a), S(b), z, GCD(S(a), S(b)) = d > 1, z d, S(c) = xy, GCD(S(a), S(b)) z S(a) S(b). : 1.6: m, S(n) = m. : S(n) [8] , S(a) x, S(b) y, S(c) w. (1-16) 1 x + 1 y = 1 w. (1-17) 1

29 Smarandache : S(a), S(b), S(c) 1, (a, b, c) (1-16)., x, y, w. (1-16) (1-17). (1-17). (1-16) x y = (x + y )w (1-18) x y x + y = w. (1-19) (1-16), x, y, w (1-18). x, y, w, (1-18) x + y,, z x + y = z, x, y, z. ( xy ) = w. (1-0) z GCD(x, y) = d, (1-0) GCD(x, y) = GCD(x, z) = GCD(y, z) = d. x = dx, y = dy, z = dz, ( dx y ) = w, (1-1) z (1-1), w, z dx y. GCD(x, z )=GCD(y, z ) = 1 GCD(x y, z ) = 1, z d d z >. z = dz, z d, (1-0) w = xy z., (1-17), x + y,, z x +y = z, GCD(x, y) = d >, z d, w = xy. (1-17). z x, y, z, (1-17). (1-16) S(n)., [3] S ( p 1 ( p 1) ) : p. S ( p 1 ( p 1) ) p + 1,

30 Smarandache, Smarandache S(n) p + 1 ( ): S ( p + 1) 6p + 1, p 7. S(n). S(n), S(n) : [4] S (F n ), : 1.7: n 3, F n = n + 1. S (F n ) 8 n + 1, : F 1 = 5, F = 17, F 3 = 57, F 4 = 65537,. n = 3 4, S (F 3 ) = , S (F 4 ) = > , n 5. F n, S(n) S (F n ) = F n = n n + 1. F n, p F n, (, p) = 1. m p, m r 1 (mod p) r. p F n, F n = n (mod p) n 1 (mod p), n+1 1 (mod p). ( [6] 10.1), m n+1, m n+1. m = d, 1 d n+1. d n, p d 1, m = n+1 m φ(p) = p 1. n+1 p 1 p = h n (1-) : (a) F n, n n , (1-), F n, p i p i = h i n n = 8 n + 1. (b) F n, 3

31 Smarandache F n = ( n ) α ( 3 n ) β F n = ( n ) α ( 3 n ) β. F n = ( n ) α ( 3 n ) β, α 4 β, S(n) S(F n ) max {S (( n ) ( α) (3, S n ) )} β = max { α ( n ), β (3 n )} 8 n + 1. F n = n + 1 = ( n ) (3 n ) = 3 n+ + n+3 + 1, n 5,. 0 n = 3 n+ + n+3 n+3 (mod n+4 ). F n = n + 1 = ( n ) ( 3 n ) = 3 3n n n+1 + n+ + n+ + 1, 0 n = 3 3n n n+1 + n+ + n+ 3 n+1 (mod n+ ).. F n = n + 1 = ( n ) 3 ( 3 n ), n + 1 ( 3 n ) 3 n+ + 1(mod n+4 ) 0 n ( 3 n ) 1 3 n+ (mod n+4 ). n+4 3 n+. F n = ( n ) α ( 3 n ) β, and α β, S(n) S(F n ) max {S (( n ) ( α) (3, S n ) )} β 4

32 Smarandache = max { α ( n ), β (3 n )} 8 n + 1. F n = n + 1 = ( n ) (3 n ), F n = n + 1 = 3 n n n = 3 n n+1 5 n+1 (mod n+3 ).. (c) F n, F n = ( n ) α Fn = ( n ) α Fn = ( 3 n ) α. F n = ( n ) α, α 4. α = 1, 3, F n = ( n ) α,. F n = ( n ) α ( 3 n ) α, α. α = 1, F n,.. S(n) : 1.6: OS(n) [1, n] S(k) k. 1.7: ES(n) [1, n] S(k) k. 1.9: ES(n) lim n OS(n). : [5],! : 1.8: n > 1, ( ) ES(n) 1 OS(n) = O. ln n 5

33 Smarandache : ES(n). n > 1, n = p α 1 1 pα pα r r n, S(n) S(n) = S (p α i i ) = m p i. m = 1, S(n) = p i, n =. M = ln n, ES(n) = k n S(k) S(k) M (1-3), kp α n αp>m, α 1 kp n p>m 1 + M <p n M <p n n ln n + kp α n αp>m, α 3 k n p 1 + n p + p n αp>m, α p n ln n + p n α> M k n S(k)=S(p α ), α kp α n αp>m, α 1 p α n αp>m, α 3 p α n αp>m, α 3 n p α + n p α + n p α 1 1. (1-3) 1 k n p α p n αp>m, 3 α<p p n p> M, α 3 n p α n p α n ln n + n + n M 1 M n ln n. (1-4) (1-3), [ ]. M p M, α(p) = p 1, α(p) M p 1. u = p α(p). S(k) M k, S(k) = S(p α ), p M S(k) p α M!,, α j=1 [ ] M p j M p 1. S(k) M k u k u 6

34 Smarandache, d(u). 1 = (1 + α(p)) = p M 1 S(k) M d u = exp ( [ M ln 1 + p 1 p M exp(y) = e y. ( [3] [5]) π(m) = p M 1 = M ln M + O p M ( [ ]) M 1 + p 1 ]), (1-5) ( M ln M ) p M = p M p M ( ) M ln p = M + O ln M ( [ ]) M ln 1 + p 1 p M [ ln (p 1 + M) ln p ln ( ln 1 + M ( 1 1 p p 1 )] ) π(m) ln(m) p M = M ln(m) ln M 1 p ln p + p M ( ) M M + O = O ln M M = ln n, (1-5) (1-6) : S(k) M c. ( ) c ln n exp ln ln n : 1 exp ES(n) = k n S(k) ( ) M. (1-6) ln M ( ) c ln n, (1-7) ln ln n n, (1-3), (1-4) (1-7) ln n ( n ) 1 = O. ln n 7

35 Smarandache OS(n) + ES(n) = n, : ( n ) OS(n) = n ES(n) = n + O. ln n ES(n) OS(n) = ( n O ln n n + O. : ) ( n ln n 1.: n, ES(n) lim n OS(n) = 0. ( ) 1 ) = O. ln n S(n), [6] PS(n) lim, PS(n) [1, n] S(n) n n, : 1.9: n > 1, ( ) PS(n) 1 = 1 + O. n ln n 1.3: n PS(n) lim n n = 1. : n PS(n). n > 1 n = p α 1 1 pα pα r r n, S(n) S(n) = S (p α i i ) = m p i. α i = 1, m = 1 S(n) = p i. α i > 1, m > 1, S(n). n PS(n) [1, n] 8

36 Smarandache S(n) = 1 S(n) n! S(n) = 1 n = 1. M = ln n, n PS(n) = (1-8) kp α n αp>m, α k n S(k)=S(p α ), α kp n p>m kp α n αp>m, α 3 S(k) M (1-8), M <p n M <p n n ln n + k n p 1 + n p + p n αp>m, α p n ln n + p n α> M p α n αp>m, α 3 p α n αp>m, α 3 n p α + n p α + n p α kp α n αp>m, α 1 k n p α p n αp>m, 3 α<p p n p> M, α 3 n p α n p α n ln n + n + n M 1 M n ln n. (1-9) (1-8), [ ]. M p M, α(p) = p 1, α(p) M p 1. u = p α(p). S(k) M k, S(k) = S(p α ), p M S(k) p α M!, α j=1 [ ] M p j M p 1. S(k) M k u k u, d(u). 1 1 = (1 + α(p)) = ( [ ]) M 1 + p 1 S(k) M d u p M p M = exp ( [ ]) M ln 1 +, (1-30) p 1 p M 9

37 Smarandache exp(y) = e y. ( [3] [5]) π(m) = p M 1 = M ( ) M ln M + O ln, M p M ( ) M ln p = M + O ln M : p M = p M ( [ ]) M ln 1 + p 1 p M [ ln (p 1 + M) ln p ln ( ln 1 + M ( 1 1 p p 1 )] ) π(m) ln(m) p M = M ln(m) ln M 1 p ln p + p M ( ) M M + O = O ln M M = ln n, (1-30) (1-31) : S(k) M c. ( ) c ln n exp ln ln n n PS(n) = exp ( ) M. (1-31) ln M ( ) c ln n, (1-3) ln ln n n, (1-8), (1-9) (1-3) ln n k n S(k)=S(p α ), α ( n ) PS(n) = n + O. ln n ( n ) 1 = O, ln n 30

38 Smarandache 1.15 Smarandache 1.8: n, Sdf(n) m m!! n, Sdf(n) = min{m : m!! = kn, m N, k N}. : 1,, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,. 1.30: Sdf(n + 1) Sdf(n). Sdf(n + 1) 1.31: Sdf(n) = k, k, n > :. Sdf(n) Sdf(n + 1) = k. 1.3: Sdf(n) k : Sdf k (n) = Sdf(Sdf(Sdf (Sdf(n)) )), Sdf k. n, Sdf(n). 1.33:. k, Sdf(n) Sdf(k + n) 1.34: n 1, Smarandache Sdf(n) Smarandache. 1.35: 1.36: 1.37: ( 1) k Sdf(k) 1. k=1 1 Sdf(n). n=1 Sdf(k) lim k θ(k), θ(k) = ln(sdf(n)). n k 31

39 Smarandache 1.39: m,n,k, Sdf(n m) = m k Sdf(n). 1.39: 1.40:. Sdf(n)! = Sdf(n!). k > 1 n > 1, Sdf(n k ) = k Sdf(n). 1.41: k > 1, Sdf(n k ) = n Sdf(k) 1.4: Sdf(n k ) = n m Sdf(m), k > 1, m, n > : Sdf(n), n Sdf(n) 1 n +, 1 n n > 1000,. 1.44: Sdf(n), Sdf(n) n 1, 1 n 1000 n0.73. n > 1000,. 1.45: Sdf(n), 1 n + 1 Sdf(n) < n 1 4, 1 n n > 1000,. 1.46: Sdf(n), 1 n Sdf(n) < n 5 4, 1 n

40 Smarandache. n > 1000,. 1.47: Smarandache Sdf(n) n=1 1 Sdf a (n), a > 0, a R. 1.48: lim n n k= n. ln Sdf(k) ln(k) 1.49: Sdf(n) r + Sdf(n) r Sdf(n) = n, r. Sdf(n) r + Sdf(n) r Sdf(n) = n, r, k. 1.50: 1.51: ( m m Sdf m k ) Sdf(m k ). k=1 n=1 k=1 1 Sdf(n). : 1 p 1 Sdf(n) p n=1, p., p, Sdf(p) = p, 1 Sdf(k) > 1 p,. k=1 p Sdf(n) 1.5:. n n=1 33

41 Smarandache Sdf(k) Sdf(p) :, >, p. k p k=1 p=1 Sdf(p) p, Sdf(p) = p,. p p=1 1.53: n, Sdf(n) 1. :., n = 1, 1. n 1, 1 n, Sdf(n) : Sdf(n) = Sdf(n + 1). :, Sdf(n), Sdf( )=, Sdf( )=. n, Sdf(n), n+1, Sdf(n+ 1).., n,. 1.55: Sdf(n) S(n). : Sdf(n) Sdf(n)! S(n). : 1.55, [7] Sdf(n),., [8] 1.55, Sdf(n) n P (n). : 1.10: n, x 1 n x Sdf(n) = x ln x ( ) x ln x ln ln x + O (ln ln x). : S(n) Sdf(n), Sdf(n)! S(n) (Sdf(n) 1)!. 34

42 Smarandache i Sdf(n) Euler [3], : i Sdf(n) i Sdf(n) 1 ln i ln S(n) i Sdf(n) 1 ln i. ln i = m ln m m + O(ln m), ln i = m ln m m + O(ln m). m ln m m + O(ln m) ln S(n) m ln m m + O(ln m), ln S(n) = m ln m m + O(ln m), (1-33) ln S(n) m = ln m 1 + O(1). (1-33) ln m ln ln S(n), m = = ln S(n) ln ln S(n) 1 + O(1) ln S(n) ln ln S(n) + O ( ln S(n) ln ln S(n) ). (1-34) m = Sdf(n) (1-34) n x Sdf(n) = n x ln S(n) ln ln S(n) + O ( n x ) ln S(n) ln. ln S(n) n x ln S(n) ln ln S(n) n x ln S(n) ln ln S(n) ln n ln ln n. n x n x ln n ln ln n = x ln x ( x ) ln ln x + O ln ln x 35

43 Smarandache x ln x ( ) x ln x ln ln x + O ln. (1-35) ln x n, n = p α 1 1 pα...pα k k n. 1 n x n A B, A [1, x] α i n i = 1,,, k. B [1, x] A, n x ln S(n) ln ln S(n) = n x n A ln S(n) ln ln S(n) + n x n B ln S(n) ln ln S(n). (1-36) S(n) n x n A ln S(n) ln ln S(n) n x n A ln x ln ln x x ln x ln ln x O ln x ln ln x ( ln x ln ln x 1 n x n A ). (1-37) n x n B ln S(n) ln ln S(n) = np x (n, p)=1 ln S(np) ln ln S(np) > np x (n, p)=1 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 1 p ln ln p + O ln p. (1-38) ln ln p p x p x Abel s p x ln p p 1 ln ln p = ln x ( ) ln x ln ln x + O ln ln x (1-39) p x ln p ln ln p x ln ln x. (1-40) 36

44 Smarandache (1-35), (1-36), (1-37), (1-38), (1-39) (1-40), n x. Sdf(n) = x ln x ln ln x + O ( ) x ln x (ln ln x). : 1.11: n x x > 1 k, (Sdf(n) P (n)) = ζ(3) 4 x 3 k ln x + i= c i x 3 ( x 3 ln i x + O ln k+1 x ), P (n) n, c i. 1.1: x > 1 k, n x (Sdf(n) S(n)) = ζ(3) 4 x 3 k ln x + i= c i x 3 ( x 3 ln i x + O ln k+1 x : [1, x] n A B : A = {n : 1 n x, P (n) > n}; B = {n : 1 n x, n / A}, P (n) n. n A, n = m P (n) P (m) < P (n). A Sdf() =. n > n A, n, Sdf(n) = P (n); n, Sdf(n) = P (n)., (Sdf(n) P (n)) n x n A = n x n A (Sdf(n) P (n)) + = (Sdf(n) P (n)) = n x n A n 1 x n 1 A 1<n x n A ). (Sdf(n 1) P (n 1)) (P (n) P (n)) 37

45 = 1<n x n A Smarandache P (n) = np x p>n p = n x n<p x n p. (1-41) Abel ( [3] 4.) ( [7] 3.): k ( ) a i x x π(x) = ln i x + O ln k+1, x i=1 a i (i = 1,,, k), a 1 = 1. n<p x n p = = x ( x ) x (n) π (n) n π(n) y π(y)dy n n x 3 k 4n 3 ln x + i= b i x 3 ln i n n 3 ln i x ( + O n x, b i. 1 = ζ(3), (1-41) (1-4) n3 n x n A n=1 (Sdf(n) P (n)) = ζ(3) 4 x 3 k ln x + i= x 3 n 3 ln k+1 x c i x 3 ( x 3 ln i x + O ln k+1 x ).(1-4) ), (1-43) c i. n n B, Sdf n ln n P (n) n. (Sdf(n) P (n)) n ln n x ln x. (1-44) n x n x n B (1-43) (1-44) n x (Sdf(n) P (n)) = n x n A 38 = ζ(3) 4 (Sdf(n) P (n)) + n x x 3 k ln x + i= n B c i x 3 ( x 3 ln i x + O ln k+1 x (Sdf(n) P (n)) ),

46 Smarandache c i S(n) P (n) = 0, n A; S(n) P (n) n, n B. [9] 1.11, n x (Sdf(n) S(n)) = n x (Sdf P (n)) + n x = ζ(3) (S(n) P (n)) n x (S(n) P (n)) (Sdf(n) S(n)) x 3 k ln x + i= c i x 3 ( x 3 ln i x + O ln k+1 x ). 1.56: ( m m Sdf m k ) Sdf(m k ). k=1 k=1 : [30], : 1.13: k 4, (m 1, m, m k ) ( k ) Sdf m i = i=1 k Sdf(m i ). i=1 1.14: k 5, (m 1, m,, m k ) ( k ) Sdf m i = i=1 k Sdf(m i ). i=1, Vinogradov, : 1.7: K > 0, K n. n = p 1 + p + p 3, p i (i = 1,, 3). 39

47 Smarandache : [8]. 1.8: k 3, n k, n = p 1 + p + + p k. : [31]. : k 4, 1.8, p, k : p = p 1 + p + + p k. Sdf(n) Sdf(p) = p. p = Sdf(p) = Sdf(p 1 + p + + p k ) = p 1 + p + + p k = Sdf(p 1 ) + + Sdf(p k ). k 4, k 1 3. p k 1 : p, p = p 1 + p + + p k 1. p = + p 1 + p + + p k 1 p = Sdf(p) = Sdf( + p 1 + p + + p k 1 ) = + p 1 + p + + p k 1 = Sdf() + Sdf(p 1 ) + + Sdf(p k ). p, (m 1, m, m k ) ( k ) Sdf m i = i=1 k Sdf(m i ). i=

48 Smarandache k 5, k 3. p, 1.8, p k : p = p 1 + p + p p k. Sdf(p ) = 3p, Sdf(p 1 p k p p) = Sdf(p ) = 3p = p 1 + p + + p k + p k = Sdf(p i ) + Sdf(p) + Sdf(p). i=1 m i = p i, i = 1,, k, m k 1 = m k = p, ( k ) Sdf m i = i=1 k Sdf(m i ). i=1 k 5, k 3 3. p, 1.8 p 4 k 3 : p 4 = p 1 + p + + p k 3. p = + + p 1 + p + + p k 3 + p. Sdf( p 1 p p k 3 p) = p = + + p 1 + p + + p k 3 + p. m i = p i, i = 1,,, k 3, m k = m k 1 =, m k = p, ( k ) k Sdf m i = Sdf(m i ). i=1 i=1 p, (m 1, m, m k ) ( k ) k Sdf m i = Sdf(m i ) i=1 i=1 41

49 Smarandache k = 4, (m 1, m, m 3, m 4 ) ( 4 ) Sdf m i = i=1 4 Sdf(m i ) i= : n, n Smarandache 1.9: n, n Smarandache SQ(n) k nk. : 1, 1,, 6, 4, 1, 70, 3, 80, 1, ,, , 360,. 1.58: : p, SQ(p) = (p 1)!,. 1.59: 1.60:,, Smarandache,., ( Scientia Magna ): (a) d(n). + n=1 1 + SQ(n) n = m=1 m d(m!) (m + 1)!, (b) SQ(n 1) SQ(n 1) (n 1)., SQ(n 1) (n 1) = (m 1)!!, (m 1)!! = 4

50 Smarandache (m 1). {SQ(n 1)}, + n=1 d(n). 1 + SQ(n 1) (n 1) = m= Smarandache p m d((m 1)!!), (m + 1)!! 1.30: p n m!. p n 0, S p (n) m 1.61: Smarandache, p, n = 0, 1,, 3,, S p (n) p n. Smarandache. :, p., p. p k S p (n), S p (n) S p (n + k 1), p Smarandache 1.31: n,, Smarandache. :, 3, 5, 7, 11, 13, 14, 16, 17, 19, 0, 3, 9, 30,. Smarandache, : 1.3: n,, Smarandache. 1.33: n,, Smarandache. Smarandache. 43

51 Smarandache 1.6: Smarandache,,. 1.63: k n, n + 1, n +,, n + k Smarandache. : k =. 3, : SPPFK(n) Smarandache n. SPPFK(n + 1) SPPFK(n). 1.5: SPPFK(n + 1) SPPFK(n). 1.65: S pp (n) n Smarandache. S pp (n) lim. n n Smarandache Smarandache 1.34: n,, Smarandache. : 1, 4, 9, 10, 16, 18, 5, 36, 40, 49. Smarandache, : 44

52 Smarandache 1.35: n,, Smarandache. 1.36: n,, Smarandache. 1.66: Smarandache 1.6: Smarandache. 1.67: k Smarandache. 1.7: k. n, n + 1, n +,, n + k 1.68: SPSFK(n) Smarandache n, SPSFK(n + 1) SPSFK(n).. 1.8: lim (SPSFK(n + 1) SPSFK(n)) n 1.69: Smarandache. 1.70: : 1.37: n, m, Smarandache m : : 1.38: n,, Smarandache

53 Smarandache 1.0 Goldbach-Smarandache 174, Goldbach : n 4. Goldbach, Smarandache : 1.39: t(n) = m m n. : 6, 10, 14, 18, 6, 30, 38, 4, 4, 54, 6, 74, 74, 90,. 1.7: : n 1.1 Vinogradov-Smarandache. Vinogradov., Smarandache : 1.40: v(n) = m m 9 n. : 9, 15, 1, 9, 39, 47, 57, 65, 71, 93, 99, 115, 19, 137, : 1.75: 3n 46

54 Smarandache 1. Smarandache-Vinogradov,. 1.41: a(k + 1) k + 1. : 0, 0, 0, 0, 1,, 4, 4, 6, 7, 9, 10, 11, 15, 17, 16, 19, 19, 3, 5,. 1.76:,, (17, 16) (43, 39) : a(k + 1) lim k k + 1 Smarandache-Goldbach : 1.4: b(k) k. : 0, 1, 1, 1,, 1,,,,, 3, 3, 3,, 3,, 4,. 1.78:. 1.79: b(k) lim k k 1.3 Smarandache-Logics 1)Smarandache Paradoxist 1.43: n Smarandache Paradoxist Smarandache. 47

55 Smarandache ) Smarandache 1.44: n Smarandache Paradoxist Smarandache Smarandache Paradoxist. : Smarandache Paradoxist Smarandache. 1.4 Smarandache-Position 1.45: x n, { (max(i)), x 10 i ; U k (x n ) = 1,., x n k =, : 0, 1, 1, 1,,,,, 0, 0,,,,,. 1.80: 1.81: k, x n n., {1,,..., 9} 1.5 Smarandache 1.46: p 1 p, p 1 < p p p 1 =, p 1 p. 1.47: p, p p + (p 1)! + 1 p + (p + 1)! + 1 p + (1-45). 48

56 Smarandache : p p +,. Wilson, (1-45). 1.8: p, : p p + { 1 (p 1)! p + } + 1 p + p + 1 p :. : [34], : : 1.8. Wilson s, p, (p 1)! 1(modp). p (p 1)! + 1. (p 1)! + 1 p (1-46). p +, (p + 1)! + 1 0(modp + ), (p 1)! p (p + 1) + 1 0(modp + ) (p 1)! + 1 0(modp + ). (p 1)! + 1 p +. (1-47) (p 1)! { 1 p + } + 1 p + p + 1 p + = (p 1)! + 1 p + (p 1)! + 1, p + 49

57 Smarandache (1-46) (1-47) { 1 (p 1)! p + } + 1 p + p + 1 p +. (p 1)! { 1 p + } + 1 p + p + 1 p + (1-48), 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, (1-48). p > 4, a (p 1)!, b (p 1)!, p = ab (p 1)! ( a = b, a (p 1)!, p (p 1)!)., { 1 (p 1)! p + } + 1 p + p + 1 p +. (p 1)! + 1 (b), p. (p 1)!. { 1 p + } + 1 p + p + 1 p + (c), { 1 (p 1)! p + } + 1 p + p + 1 p (p + 1)! 1, p + p + = (p 1)! + 1 p + (p + 1)! p p + (p 1)! (p + 1)! + 1, 1. p p + p = (p 1)! p + (p + 1)! + 1 p p

58 Smarandache (p 1)! + 1 p + (p + 1)! + 1 p + = = = (p 1)! (p + 1)! p p + p (p 1)! + 1 (p + 1)! + p p p + (p 1)! (p + 1)! + p p p + 1 p +. (1-49) p p +, (1-49) 1 p +. p + p > 1, (1-49) 1 p. p p + p > 1, (1-49) 1 p + 1 p Smarandache Smarandache : k, Diophantine y = x 1 x x k + 1, y x i, i = 1,,, k.,.,.,, p m = p 1 p p n + 1, p 1 =, p 1 < p < < p n, m > n. : + 1 = 3, = 7, 51

59 Smarandache = 31, = 11, = 311,,. 1.84: n p m = p 1 p p n + 1 (1-50), p i, i = 1,,, n, m > n : m = n, m = n, m = n(n + 1), 1.86: y = x 1 x x k + 1 (1-51), k x 1 x x k. k,. 1.7 Smarandache Smarandache : ap n + b, (a, b) = 1 p n n. 1.9:. 1.87: a n + b, (a, b) = 1, a { 1, 0, 1}. 5 Dirichlet.

60 Smarandache 1.10: a + b,. p,, p. p. 1.88: n n + 1 n n 1, n = 1,, 3,. : n =, n n 1. n, n n 1 ; n, n n 1 n n 1 = (n n + 1)(n n 1). 1.8 Smarandache Counter. 1.48: Smarandache CounterC(a, b) a b Smarandache C(1, p n ), C(1, n!), C(i, n i ), p n n. Smarandache Counter, : p n n. 1 i 9 C(i, m) = [log 10 m] + 1; 1 C(1, p n ) + C(3, p n ) + C(7, p n ) + C(9, p n ), 1.9 Smarandache C(n) 1.49: C(n) m n n C m n = n! m! (n m)!. C(n) = max{m : m n, n C m n }, C(1) = C() = 1. 53

61 Smarandache Smarandache C(n),., ( ): (a) k >, n n C(n) k. (b) n 4, ( C(n) = n + O exp exp(y) = e y, c 1 > 0. (c) N 4, : ( c1 ln n ln ln n )), C(n) = 1 N + O (N ln N). n N 1.30 Smarandache G(n) Smarandache G(n) : 1.50: G(n) m n min{m : n m φ(k), G(n) = k=1 m φ(k), m N}, φ(n). k=1 G(n), [36] : 1.15: p, G(p) = min{p, q(p, 1)}; G(p ) = q(p, ), q(p, ) < p ; G(p ) = p, q(p, 1) < p < q(p, ); 54 G(p ) = q(p, 1), p < q(p, 1) < p ; G(p ) = p, q(p, 1) > p,

62 Smarandache q(p, i) {np + 1} i : G(n) Smarandache, Dirichlet n=1 G(n) n 1.17: k, (m 1, m,, m k ), G(m 1 m m k ) G(m 1 )G(m ) G(m k ). : G(p) = m, G(n) p m φ(k), p k=1 m. m m = q α 1 1 qα qα r r 1)q α 1 (q 1) q α r 1 r 1 i r. p q α i 1 i s φ(k), 0 < s < k=1, φ(m) = q α (q 1 (q r 1), G(n) p φ(q α i i ), p q i 1. : (i) p q i 1, m = q i = lp + 1 {kp + 1}. (ii) p q α i 1 i, α i =, m = q i = p. (i) (ii) G(p) = min{p, lp+1}, lp+1 {kp+ 1}., G(p ) n > 1, n n = p α 1 1 pα pα r r, G(p α i i ) = m i, 1 i r, m = max{m 1, m,, m r }, G(n) p α i i φ(1)φ() φ(m i ), 1 i r. p α i i (p i, p j ) = 1, i j, n = p α 1 1 pα pα r r, G(n) = m = max{g(p α 1 1 ), G(pα ),, G(pα r G(n) Smarandache n=1 G(n) n > p G(p) p > p 1 p =. φ(1)φ() φ(m). φ(1)φ() φ(m). r ). 55

63 Dirichlet Smarandache n=1 G(n) n., m 1 m, G(m 1 m ) G(m 1 )G(m ). G(m 1 ) = u, G(m ) = v, G(n) u v m 1 φ(i), m φ(i). i=1, u v, uv u uv φ(k) = φ(k) φ(k) = k=1 k=1 k=u+1 i=1 u φ(k) φ(u+1) φ(v) φ(v) φ(uv). k=1 φ() φ(v), φ(3) φ(3v),, φ(u) φ(uv), φ(u+1) φ(u+1),, φ(v) φ(v), v φ(i) i=1 u φ(i) i=1 uv k=u+1 v φ(i) i=1 m 1 m uv k=1 φ(k), uv k=1 φ(k). G(n) G(m 1 m ) uv, k 3, φ(k). G(m 1 m ) G(m 1 )G(m ). G(m 1 m m k ) = G(m 1 (m m k )) G(m 1 )G(m m k ) G(m 1 )G(m )G(m 3 m k ) G(m 1 )G(m ) G(m k ). 56

64 Smarandache 1.31 Smarandache : {a n } n N : (a) i N, j, k N, i j k, a i a j (moda k ). (b) i N, j, k N, i j k, a j a k (moda i ). 1.89: Smarandache, {a n } n N i N, j, k N, i j k, a i a j (moda k ). : a 1 a 5 (moda 3 ), a 3 a 7 (moda 5 ), a 5 a 9 (moda 7 ),, a a 6 (moda 4 ), a 4 a 8 (moda 6 ), a 6 a 10 (moda 8 ),. {a n } n N. 1.3 Smarandache 1.90: Smarandache :. :,. : { 1, x ; F (x) = 0, x. G(x) = {x : G(x) = 0, x }. F (x) = lim cos (n!x) n 57

65 Smarandache a, b Z. G(x) = ax + b, 1.33 Smarandache 3 1.5: Smarandache : 1, 1, 1, 13, 31, 31, 134, 341, 341, 413, 1345,. 1.91: Smarandache. 1.9: 1.11: Smarandache : p n n. d n = p n+1 p n, 1.93: d n. 1.94: d n n! n n. 1.95: d n n d n = k, k N. 1.96: d n n i p n+i p n = k, k N. 1.97: d n. p n lim n n log n = Smarandache : k, n i N, k < n i, : n 0 = n, n i+1 = max{p : p n i k, p } 58

66 Smarandache, n 0 = 10, n 1 = 10 3, n = 7, n 3 = 5, n 4 = 3 1, i 5, n i. n 0 1,. n 1 n 0, n n 1,. n 0, n 0. n 0, n : k, n i N, k < n i, : n 0 = n, n i+1 = max{p : p n i k, p }.. : n 0 = 0, n 1 = 5, n = 5, n 3 = 5,.,, n 1 n 0 n i = n 1, i. 1.99: : n 0 = n, n i+1 = max{p : p n i + k, p }, k, n N, 1 k n : : n 0 = n, n i+1 = max{p : p n i k, p }, k, n N, 1 k n Smarandache :, a Q \ { 1, 0, 1}. xa 1 x + 1 x ax = a (1-5) 59

67 Smarandache : [37], : : a Q \ { 1, 0, 1}, (1-5) x = 1.19: R. a R \ { 1, 0, 1}, x a 1 x + 1 x ax = a x = 1; a > 0, x = 1. : Rolle s a > 1 (1-5)., x (1-5), x > 0. u + v u v x a 1 1 x + x ax x a 1 x 1 x ax = a x+ 1 x a, x = 1 (1-5). a > 1 x = 1. 0 < a < 1. x 0 (1-5), (1-5) x 0 > 0. x 0 = 1. x 0 1, 0 < x 0 < 1, 1 x 0 > 1. f(x) : f(x) = xa 1 1 x + x ax a. ] ) f(x) [x 0, 1x0, (x 0, 1x0 f(x 0 ) = f( 1 ) = f(1) = 0. Rolle s, x 0 ) f (x) (x 0, 1x0, f (x) ). f(x) x (x 0, 1x0, 60 f (x) = a 1 x 1 x a 1 x ln a 1 x ax + 1 x ax ln a

68 Smarandache f (x) = 1 x 3 a 1 x ln a + x 3 a 1 x 1 x a 1 x ln a 1 x a 1 x ln a + 1 x a 1 x ln a = 1 x 3 a 1 x ln a + x 3 a 1 x + x a 1 x ln 1 a + 1 x a 1 x ln a > 0, 0 < a < 1, ln 1 a > 0, ( x 0, 1 x 0 ) f (x). 0 < a < 1. a < 0 a 1, (1-5) x, x,., (1-5) : a = a = x a 1 x 1 x ax = x ( 1) 1 x a 1 x 1 x ( 1)x a x = x a 1 x + 1 x a x : xb 1 (n 1)π x cos{ } + 1 x x bx cos{(n 1)πx} = b Smarandache : n, d(n) n. Smarandache k. d(d( d(n) )) = d k (n) = (1-53) : n, d(n) =. d() =, d k (n) = k =. (1-53) k =. 61

69 Smarandache 1.38 Smarandache : {a n }, N(n) n. Smarandache : 1.104: m, {a n } m, {a n } 0, 1, m, 3 m,. n, k N k (n). 1.1: lim n k log m (log n) = Smarandache 9 Smarandache : 1.13: k N, p, q, x, y > 1, x p y q = k (1-54). :, x 3 y 5 = 7 : (x )(x + x + 4) = (y 1)(y 4 + y 3 + y + y + 1). (1-54) Smarandache : r {1,, 3,, r} n x, y, z, x y z, x + y = z. 6

70 Smarandache :., 10, 10 :{x 0(mod }, {x 1(mod 10)},, 1 11 = 11, 5 15 = 75, 6 19 = 96, 10 0 = 400. r = 74 = : n, : x y z, x + y = z, : 1.106, n,., n = 5, {0, 1,, 3, 4, }, 5 10 = 50, 6 11 = 66. r = 49 = , 10, = 30, r = 9 = n Smarandache : 1 a 1 a a n. Smarandache : :. a n n=1 : Kenichiro Kashihara, a n = n n +,. : 1,, 4, 7, 11, 16,, 9, 37, 46, 56, 67, 79,. 63

71 Smarandache (1, 4, 7), (, 9, 56) (7, 37, 67), a n = n n +. :. : a 1 = 1, a n = a n 1, a n+1 = a n +. : 1,, 4, 8, 10, 0,, 44, 46, 9, 94,., b n = n 1. b n = n 1 : 1 =. b n n=1 1,, 4, 8, 16, 3, 64,, 1.109: Smarandache a n = n 1 a 1 = 1, a n+1 = a 1 a n + 1. : a 1 = 1, a n+1 = a 1 a n + 1 S n = k=1 (1-55). n=1 1 a k, S n + 1 a n =. (1-55) 1 a n Smarandache 1 =. 1.57: e p (n) n p. 64

72 Smarandache, p = 3, : 0, 0, 1, 0, 0, 1, 0, 0,, : e p (n) : e p (n) = 1, p n, p n. 1 p 1 p. e p (n) k N ( 1 k p k 1 ) p k+1 = k N k ( ) p 1 p k+1 = (p 1) k N = p 1 (p 1) 1 = p 1. k p k+1, p e p (n) : e p (n), e q (n), e m (n) m = p q. : (p, q) = 1 (p, q) 1, min{e p (n), e q (n)} = e pq (n). min{e p (n), e q (n), } = e m (n) Smarandache 13 Smarandache : 1.11: x y [x] = y (1-56) 65

73 Smarandache, [x] x , Smarandache. : (a) y R Q, (b) y = m n Q Z. : Smarandache y 1, x = (y + 1) 1 y. (1-57), (1-57). y > 0, y + 1 < (1 + 1) y = y. (1-57) (1-56), x y [x] = (y + 1) y y (y + 1) 1 y = y = y. (1-58) (1-58) 0 < y R, (1-56) x = (y + 1) 1 y. y < 0,, (1-56), : x y [x] y = y : x y [x] y = x : x[y] [x]y = x y : x [y] y [x] = x y. : 1.115, ( ): 1.0: x[y] [x]y = x y (I) x = y = a, a R, R ; 66

74 Smarandache (II) { x = ( 1) i + r x, y = ( 1) i + r y. r x, r y [0, 1) r x > r y i = 0, r x < r y i = 1; (III) { x = mq + ( 1) i + r x, [ ] y = mp + ( 1) i p q 1 + ( 1) i r x. p, q, m Z, (p, q) = 1, r x [0, 1), Z, p < q, i = 0; p > q, i = 1. : Gauss [x]. (1) x = y = a, a R. () x y r x, r y x y, r x, r y [0, 1) r x r y. r x = r y = r 0 r < 1, x = [x] + r, y = [y] + r. ([x] + r)[y] [x]([y] + r) = [x] [y]. r([y] [x]) = [x] [y]. r = 0 [x] = [y], x = y, x y. r > 0 r([y] [x]) = [y] [x]. ([y] [x])(1 r) = 0. r 1 [y] = [x], x = y, x y r x r y. x = [x] + r x, y = [y] + r y r x, r y [0, 1). (a) x > y, x[y] [x]y = x y x([y] 1) = y([x] 1). x = 1 + r x, y = 1 + r y, 0 r y < r x < 1. 67

75 Smarandache y 0 [y] 1 0, x y = q, (p, q) = 1, q > p, p, q Z. p [x] 1 [y] 1 = q p. q [x] 1, p [y] 1. [x] = mq + 1, [y] = np + 1, m, n Z, [x] 1 [y] 1 = q p = mq np, m = n, x = mq r x, y = mp r y. x y = mq r x = q mp r y p, p(mq r x ) = q(mp r y ), p(1 + r x ) = q(1 + r y ), r y = p q (1 + r x) 1. x = mq r x, y = mp + p q (1 + r x). (p, q) = 1, q > p, p, q, m Z, r x [0, 1). x > y, : x = 1 + r x, y = 1 + r y, 0 r y < r x < 1, x = mq r x, y = mp + p q (1 + r x). (p, q) = 1, q > p, p, q, m Z, r x [0, 1). (II), (III) i = 0. 68

76 Smarandache (b) x < y, x[y] [x]y = y x x([y] + 1) = y([x] + 1). x = 1 + r x, y = 1 + r y 0 r x < r y < 1. y 0 [y] + 1 0, x y = q, (p, q) = 1, p > q, p, q Z, p [x] + 1 [y] + 1 = q p. q [x] + 1, p [y] + 1. [x] = mq 1, [y] = mp 1, m Z, x = mq 1 + r x, y = mp 1 + r y. x y = mq 1 + r x mp 1 + r y = q p, p(mq 1 + r x ) = q(mp 1 + r y ), p(1 r x ) = q(1 r y ), r y = 1 p q (1 r x). x = mq 1 + r x, y = mp p q (1 r x). (p, q) = 1, p > q, p, q, m Z, r x [0, 1). x < y, : x = 1 + r x, y = 1 + r y, 0 r x < r y < 1, x = mq 1 + r x, y = mp p q (1 r x), 69

77 Smarandache (p, q) = 1, p > q, p, q, m Z, r x [0, 1). (II), (III) i = 1. (1)() Smarandache : n, Smarandache SPB(n) n a i p i = n (a n a n 1 a 0 ). p 0 = 1, i 1, p i i i=0, a i = { 1, i p i ; 0, i. SPB() = 10 = + 0, SPB(3) = 11 = + 1. SPB(14) = = 13+1, = = : 0, 1, 10, 100, 101, 1000, 1001, 10000, 10001, 10010, 10100, :, SPB(n). Smarandache, : : n > 1, p n p n+1 p n n < p n , n, p n p n+1 p n n < p n+1, n = p n + r 1. r 1, p m p m+1 p m n < p m+1, r 1 = p m +r, m < n., j r j = 0. n, SPB(n). 70

78 Smarandache 1.118: n : 1.117, p k < n <p k+1, SPB(n) k+1. n :

79 Smarandache Smarandache.1 Kenichiro Kashihara Smarandache, Smarandache Z(n). Smarandache., C.Ashbacher Smarandache., Smarandache. Smarandache :.1: n, Smarandache S(n) m n m!, S(n) = min{m : n m!, m N}. Smarandache Smarandache,,, :.: n, Smarandache Z(n) m m n k. 7 k=1 1 n 60, Z(n) : n Z(n) n Z(n) n Z(n) n Z(n) n Z(n) n Z(n)

80 Smarandache :.3: U(n) : U(1) = 1. n > 1 n = p α 1 1 pα pα s s n, U(n) = max{α 1 p 1, α p,, α s p s }. Smarandache..4: S c (n) m y n!, y 1 y m, S c (n) = max {m : y n!, 1 y m, m + 1 n!}..5: n, Smarandache Zw(n) m n m n, Zw(n) = min{m : m N, n m n }..6: Euler φ(n) n n.. Smarandache Smarandache Z(n),...1: n, Smarandache Z(n) 1. :. n = 1, Z(n) = 1...: n, Z(n) < n. : : Z() = 3, Z(4) = 7, Z(8) = : p 3, Z(p) = p 1. : Z(p) = m, m. m m(m + 1) p m k = k=1 m(m + 1), 73

81 Smarandache., p m m + 1, p = m + 1 p 1 = m, p., p =, Z() = 3...4: p 3 k N, Z(p k ) = p k 1. p =, Z( k ) = k+1 1. : Z(p k ) = m, m. m k = k=1 m(m + 1), m p k m(m + 1)., p k m m + 1, p k = m + 1 p k 1 = m, p., p =, Z() = 3...5: n, Z(n) = max{z(m), m n}. : n, : Z(n) max{z(m), m n}. Z(n) = p, Z(m) = q, m n. q > p, n p(p + 1) q(q + 1), m...6: (1) Z(n), Z(m + n) Z(n) + Z(m). () Z(n), Z(m n) Z(n) Z(m). : : Z( + 3) = Z(5) = 4 5 = Z(3) + Z(), Z( 3) = Z(6) = 3 3 = Z() Z(3). n 1..7: lim n Z(k). k=1 74

82 Smarandache :, Z(n) n k=1 1 Z(k) > n p=3 1 Z(p) = 3 p n 1 p 1 > 3 p n 1 p., p 1 n., p k=1 1 Z(k)...8: lim n n k=1 Z(k) k. :, Z(n) n k=1 Z(k) k > n p=3 Z(p) p = 3 p n p 1 p > 3 p n 1 p. 3 p n 1 n., p k=1 Z(k) k...9: m 1, n 1, Z(n) = m. :, n = m(m + 1)..3 Smarandache Smarandache, Smarandache,..3.1: Z (n)=z(z(n)),, Z k (n)=z(z( Z(n) )), Z k. k, m N, Z k (n) = m. : Z k (n) = m...9, n 0 Z(n 0 ) = m...9, n 1, Z(n 1 ) = Z(n 0 )...9 k, Z k (n k 1 ) = m. 75

83 Smarandache.3.: : (1) Z(n + 1) Z(n), Z(n + 1) (). Z(n) Z(n) = p, Z(n + 1) = q, : p(p + 1) q(q + 1) n, n + 1. (a) n, n p, n + 1 q. (b) n, n 1 p, n q. (c) n = p, p, Z(n 1) n 3, Z(n) = n 1, Z(n + 1) n + 1.,. : [38], [40].3. (1)., :.3.1: M, n Z(n + 1) Z(n) > M Z(n + 1) Z(n) > M. Z(n + 1), Z(n + 1) Z(n) Z(n) :..3.1: k h (h, k) = 1, nk+h, n = 0, 1,, 3,. : Dirichlet, [3]. :., M, m m > M. ( m+1, m + 1 ) = 1, Dirichlet, :. 76 m+1 k + m + 1, k = 0, 1,,

84 Smarandache, k 0 m+1 k 0 + m + 1 = P. P, Z(n) Z(P ) = P 1 = m+1 k 0 + m, Z(P 1) = Z( m+1 k 0 + m ) = Z( m ( m+1 k 0 + 1)). m+1 k 0 i=1 i = m+1 k 0 ( m+1 k 0 + 1) m+1 k 0 m ( m+1 k 0 + m ) i, Z(P ) Z(P 1). i=1 Z(P 1) m+1 k 0. Z(P ) Z(P 1) m+1 k 0 + m m+1 > m > M. k 0 Z(P ) Z(P 1) Z(P ) Z(P 1) m+1 k 0 + m m+1 k 0 = m+1 k 0 ( m 1) + m > m > M. Z(P ) Z(P 1). m m > M, n Z(n + 1) Z(n + 1) Z(n).. Z(n).3. (1) :.3.: M, s Z(s) Z(s + 1) > M. 77

85 Smarandache : M, α s = α > M + 1. s + 1, Z(s) = Z( α ) = α+1 1. Z(s + 1) s = α. Z(s) Z(s + 1) ( α+1 1) α = α+1 1 > M = M. s Z(s) Z(s + 1) > M...3.3: (1) Z(m + n) Z(m), Z(n), () Z(mn) Z(m), Z(n)..3.4: (1) Z(n) = Z(n + 1), () Z(n) Z(n + 1), (3) Z(n + 1) Z(n). : Z(n) 50, () : Z(6) Z(7), Z() Z(3), Z(8) Z(9), Z(30) Z(31), Z(46) Z(47). (3) : Z(10) Z(9), Z(18) Z(17), Z(6) Z(5), Z(4) Z(41), Z(50) Z(49)..3.4 (1), [40], :.3.3: Z(n) = Z(n + 1). : Z(n) = Z(n + 1) = m, Z(n) : 78 n m(m + 1), n + 1 m(m + 1).

86 Smarandache (n, n + 1) = 1, n(n + 1) m(m + 1) n(n + 1) m(m + 1)., n n + 1, (a) n, Z(n) = m n 1 < n. (b) n + 1, Z(n + 1) = m < n. (a) (b) n < m. (-1) m < n. (-) (-1) (-), n < m < n,...3.5: (1) Z(n) + Z(n + 1) = Z(n + ), () Z(n) = Z(n + 1) + Z(n + ), (3) Z(n) Z(n + 1) = Z(n + ), (4) Z(n) = Z(n + 1) Z(n + ), (5) Z(n + 1) = Z(n) + Z(n + ), (6) Z(n + 1) Z(n + 1) = Z(n) Z(n + ),.3.6: m, n Z(n) = m : 1 m 18, : m n m n , , 33, 66 3, , 6, 39,78 4 5, , , , 4, 30, 40, 60, , 14, , 34, 68, , 1, 18, , ,

87 Smarandache. Kenichiro Kashihara m, Z(n) = m. : (a) m, Z(n) = m. (b) ( Z(n) = m n) m. m = 7, Z(4) = 7. n = 4. (c) C(m) Z(n) = m n.. lim m m k=1 C(k) k m.3.7: (1) Z(n), Z(n + 1), Z(n + ), Z(n + 3) n. () Z(n), Z(n + 1), Z(n + ), Z(n + 3) n. Z(n) 35, : Z(6) = 3 < Z(7) = 6 < Z(8) = 15, : Z(1) = 6 < Z() = 11 < Z(3) =, Z(30) = 15 < Z(31) = 30 < Z(3) = 63. Z(8) = 15 > Z(9) = 8 > Z(10) = 4, Z(13) = 1 > Z(14) = 7 > Z(15) = 5, Z(16) = 31 > Z(17) = 16 > Z(18) = 8. Kenichiro Kashihara, : 80

88 Smarandache.3.8:.3.9:.3.10: Z(n) Smarandache S(n). : (A) Z(n) = S(n) ; (B) Z(n) + 1 = S(n). : [41] (A) (B).3.4: n > 1, Z(n) = S(n) n = p m, p m p m p + 1 m > 1. : n = 1, Z(n) = S(n). n =, 3, 4, 5, Z(n) = S(n). n 6 Z(n) = S(n), Z(n) = S(n) = k. Z(n) S(n) k n : n k(k + 1), n k!. (-3) (-3) k+1. k+1, p(p 1) k + 1 = p, n (n, p) = 1, n p 1. p (p 1)(p ) n i =. k = p 1 i=1 k(k + 1) n! (n, p) > 1, p, p n. n k! p k!., p = k + 1, p (p 1)!. (-3) k + 1. (-3) k k. k k + 1, k, k, 81

89 Smarandache ( k = a b a > 1, b > 1 a b. k, k + 1 ) = 1, k = a b (k 1)!, k + 1 k(k + 1) (k 1)! (k 1)!. n k(k + 1) n (k 1)!. k n k!. k, k = p α α. k p 3, p, p,, p α 1 k 1 (k 1)!, k(k + 1) n n (k 1)!. k! k! k k = p, n p(p + 1) p! n. n p + 1, S(n) < p; n = p Z(n) S(n). n = p m, m p n = p m, m p Z(n) = S(n). S(pm) = S(p) = p. m p 1 i = i=1 p(p 1), m p + 1! Z(pm) = p, Z(pm) = S(pm)., k Z(n) = S(n) = k.. k = m Z(n) = S(n) = k = m, k(k + 1) Z(n) S(n) n = m(m+1) (m)!. m 1 m + 1, (n, m + 1) = 1, n i = m(m 1), m n m(m + 1)! (n, m + 1) > 1, p = m + 1 n, n (m)! p = m + 1 (m)!,! m + 1 m, n (m 1)!, m n (m)!! m p k = p. n p(p + 1)! n (p)! S(n) = Z(n) = p. n p(p + 1)! k p(p + 1), S(k) = p i=1 8

90 Smarandache.3.5: n, Z(n) + 1 = S(n) n = p m, p m p 1. m p 1. :.3.4,. n Z(n) + 1 = S(n), Z(n) + 1 = S(n) = k. Z(n) S(n) k k(k 1) n, n k!. (-4) (-4) k! n (k 1)!, k p(p 1) n k!. k = p. n ( ) p 1 S < p, n = p m, m p 1. n = p m, m p 1, n Z(n) + 1 = S(n). (-4) k = m, k 1 = m 1, n Z(n) + 1 = S(n) = m. Z(n) + 1 = S(n) n = p m, m p (A) (B). [1, 100] Z(n) = S(n) 9 n = 1, 6, 14, 15,, 8, 33, 66, 91. (B), Z(n) + 1 = S(n) [1, 50] 19 n = 3, 5, 7, 10, 11, 13, 17, 19, 1, 3, 6, 9, 31, 34, 37, 39, 41, 43, : Z(n) Smarandache U(n). : (A) Z(n) = U(n) ; (B) Z(n) + 1 = U(n). : [4] Z(n) = U(n) Z(n) + 1 = U(n),, : 83

91 Smarandache.3.6: n > 1 Z(n) = U(n) n = p m, p, m p m p + 1 m > 1. :, n = 1, Z(n) = U(n) = 1. n =, 3, 4, 5, n Z(n) = U(n). n 6 Z(n) = U(n), n = p α 1 1 pα pα s s n, U(n) = U (p α ) = αp. Z(n) U(n) αp n : n αp(αp + 1), p α n. (-5) (-5) α = 1. α > 1, p α n p α αp(αp + 1). (-6) (p, αp + 1) = 1, p α 1 α. p (-6), p α 1 > α, p α 1 α. p =, α =. (-6) = 10,! (-5) α = 1 p. n = p m. (-5) p(p + 1) p m, m p + 1. m 1. n = p, Z(p) = p 1, U(p) = p Z(n) = U(n)! n = p m, m p + 1 1, Z(n) = p, U(n) = p, Z(n) = U(n). n > 1 Z(n) = U(n) n = p m, m p : n Z(n) + 1 = U(n) n = p m, p m p 1. m p 1. 84

92 Smarandache : n = 1 Z(n) + 1 = U(n). n > Z(n) + 1 = U(n), U(n) = U (p α ) = αp. Z(n) + 1 = U(n) Z(n) = αp 1. Z(n) U(n) n αp(αp 1), p α n. (-7) (p, αp 1) = 1, (-7) p α 1 α..3.5 α = 1 p. n = p m. (-7) m p 1 1. n = p m, m p, n Z(n) + 1 = U(n). Z(n) + 1 = U(n) n = p m, m p 1.., Z(n) = U(n) Z(n) + 1 = U(n).! [1, 100], Z(n) = U(n) 9, n = 1, 6, 14, 15,, 8, 33, 66, 91. Z(n) + 1 = U(n) [1, 50] 19, n = 3, 5, 7, 10, 11, 13, 17, 19, 1, 3, 6, 9, 31, 34, 37, 39, 41, 43, : n S c (n) + Z(n) = n. (-8) : n = 1 (-8), n = 3 (-8). p 5 p α +, n = p α (-8)., Z(p α ) = p α 1, S c (p α ) = p α + 1. S c (p α ) + Z(p α ) = p α p α 1 =. n = p α (-8)., n = 1, 5, 11, 17, 9, 41 (-8)., [43] (-8), :.3.1: n, n = 1, 3 α p β+1, α 3 α +, p p 5, β p β : n Smarandache Z(n) n. 85

93 Smarandache :. n > 1 a (a, n) = 1. Euler a φ(n) 1 (mod n), φ(n) Euler, φ(n) n n. (a, n) = 1, m a m 1 (mod n). a m 1 (mod n) m. m = φ(n), a n. n., n =, 4, p α, p α, p, α,. [44] Z(n).3.13, :.3.8: n, Smarandache Z(n) n n =, 3, 4., :.3.3: m > 1, m m =, 4, p α, p α, p, α. : [5] [9] 4. :. Z() = 3 Z(4) = 7 4. n = p α, p. Z(n) n = p α Z(n) Z(n) = p α 1, p α 1 n = p α Z (n) = (p α 1) 1 (mod p α ) p α 1 n = p α = φ (p α ) = p α 1 (p 1). α = 1, p 1 =. n = + 1 = 3. n = p α (p ) Z(n) n n = p = 3. n = p α Z(n) C) (1) p α 3 (mod 4), Z(n) Z(n) = p α. (p α, p α ) = p α > 1, Z(n) = p α n = p α. () p α 1 (mod 4), Z(n) Z(n) = p α 1. (p α 1, p α ) = > 1, Z(n) = p α 1 n = p α. 86

94 Smarandache n Z(n) n n =, 3, : Z(n), n, n = m(m + 1), Z(n) = m < n. n, n = α, Z(n) = α+1 1 = n 1. Z(n),. Z(n), n x ln(z(n)), n x n x 1 Z(n) : [45] ln Z(n), :.3.9: x > 1, ln Z(n) = x ln x + O(x). n x :.3.4: x > 1, p x ln p p p x p x. = ln x + O(1), : [3] :.3.9., n > 1, n n(n 1), Z(n) Z(n) n 1. ln Z(n) ln(n 1) x ln x + O(x). (-9) n x n x 87

95 Smarandache A [1, x] square-full n( p n, p n). [46] n x n A n x ln Z(n) = n x n A ln Z(n) + n x n/ A ln Z(n). (-10) 1 = ζ( 3 ) ζ(3) x ζ( ) ( )) ζ() x O x 1 6 exp ( C log 3 5 x(log log x) 1 5, C > 0. ln Z(n) ln(n), ln Z(n) x ln x. (-11) n x n A n / A, n = 1 p p n p n..3.4, ln Z(n) = ln Z(np) ln(p 1) p x n x n/ A np x (n, p)=1 = p x = x [ x p x p + O(1) p x ln p p x p x n x p (n, p)=1 ] ln(p 1) ln p p + O(x) = x ln x + O(x). (-1) (-9), (-10), (-11) (-1) ln Z(n) = x ln x + O(x).. 88 n x Smarandache Z(n) Zw(n) :.3.15: Zw(Z(n)) Z(Zw(n)) = 0 (-13)

96 Smarandache, Z(n) Smarandache..3.16: Zw(Z(n)) Z(Zw(n)) < : Zw(Z(n)) Z(Zw(n)) > 0. : : [47],.3.10: Zw(Z(n)) Z(Zw(n)) = : Zw(Z(n)) Z(Zw(n)) < 0..3.: Zw(Z(n)) Z(Zw(n)) > 0., [48] Smarandache Z(n) :.3.5: p >, Z(p) = p : Z(n) = n 1. n = p m, p, m,.3.7: n = m, m, Z(n) = n : n, n, { n Z(n) = 1, 4 ( n 1); n, 4 ( n + 1)..3.9: n, n 3, 3 { n Z(n) = 3 1, 3 ( n 3 1); n 3, 3 ( n 3 + 1). 89

97 Smarandache :, n ( p ). n = 1, Zw(Z(1)) Z(Zw(1)) = 0, n > 1 : (1) n = p >..3.5, Zw(Z(n)) = Zw(Z(p)) = Zw(p 1), Z(Zw(n)) = Z(Zw(p)) = Z(p) = p 1., Zw(p 1) = p 1. p 1,. () n = p m, p >, m N m > Zw(Z(n)) = Zw(Z(p m )) = Zw(p m 1), Z(Zw(n)) = Z(Zw(p m )) = Z(p) = p 1. (-13), Zw(p m 1) = p 1. p = 3, m =, n = 9. (3) n = m, m N..3.7 Zw(Z(n)) = Zw(Z( m )) = Zw( m+1 1), Z(Zw(n)) = Z(Zw( m )) = Z() = 3. (-13), Zw( m+1 1) = 3. m = 1, n =. (4) n = p 1 p p k, p i > (i = 1,, k)..3.8 Z(n) = Zw(Z(n)) = { { p 1 p p k 1, 4 p 1 p p k 1; p 1 p p k, 4 p 1 p p k + 1. Zw(p 1 p p k 1), 4 p 1 p p k 1; p 1 p p k, 4 p 1 p p k + 1. Z(Zw(n)) = Z(p 1 p p k ) 90 { p 1 p p k 1, 4 p 1 p p k 1; p 1 p p k, 4 p 1 p p k + 1.

98 Smarandache n = p 1 p p k 4 p 1 p p k + 1, (-15). (5) n = 3p, p Z(n) = Z(3p) { p 1, 3 p 1; p, 3 p + 1. Zw(n) = Zw(3p) = 3p. Zw(Z(n)) = { Zw(p 1), 3 p 1; Zw(p) = p, 3 p + 1. Z(Zw(n)) = Z(3p) { p 1, 3 p 1; p, 3 p + 1. n = 3p p 5, (-13). p 3 p + 1 (-13). 1-5, (-13) : (a) 1 : n = p > p 1, Zw(Z(n)) = Zw(p 1) < p 1 = Z(Zw(n)), Zw(Z(n)) Z(Zw(n)) < 0. 4 : n = p 1 p p k 4 p 1 p p k 1, p 1 p p k 1 = t, t N. Zw(Z(n)) = Zw(p 1 p p k 1) = Zw( t) < t. Z(Zw(n)) = Z(Zw(p 1 p p k )) = t. Zw(Z(n)) Z(Zw(n)) < (b) 3 : n = m m > 1, Zw(Z(n)) = Zw(Z( m )) = Zw( m+1 1) > 3, Z(Zw(n)) = Z(Zw( m )) = Z() = 3. 91

99 Smarandache Zw(Z(n)) Z(Zw(n)) > : n Z(k) =. k=1 n(n + 1) (-14) : [49], :.3.11: n, n = 1, : n, k n Z(k) n + 9n : : n = m, Z (4k) (-15) k n Z(k) = k m Z ((k 1)) + k m Z ((k 1)) k 1, k, Z ((k 1)) k ; k, Z ((k 1)) Z() + k m 1<k m (k 1) = n +. (-16) 4 [ ] m + 1 u = m + 1. Z (8k)). (-17) k m Z (4k) k u Z (4(k 1)) + k u k 1 > 4, : k 1(mod 4), Z (4(k 1)) k ; k 0(mod 4), Z (4(k 1)) k 1; k (mod 4), Z (4(k 1)) 3(k 1); k 3(mod 4), Z (4(k 1)) 3(k 1) 1. Z (4(k 1)) Z(4) + Z(1) + 3(k 1) 9 k u 3 k u

100 Smarandache = k u 3(k 1) Z(n) 4n 1 k u Z (8k)) k u (-17), (-18) (-19) = 3(u + 1) 3 + (16k 1) 8u(u + 1) u (m + 1) + Z (4k) Z (4(k 1)) + k m k u k u 11 4 (m + 7m ) + (-15), (-16) (-0) k n Z(k) = k m Z ((k 1)) + k m 3(m + 1). (-18) 4 7(m + 1). (-19) Z (8k)) = n + 9n (-0) Z (4k) n + 9n n. n, n = m + 1, Z((m + 1)) m + 1, n, Z(k) = k n k m+1. Z (k) = Z((m + 1)) + < n + 9n k m Z (k) : n 64 n Z(k) < k=1 n(n + 1). (-1) 93

101 Smarandache n 64, u = [ ] n+1, Z(k + 1) k, Z (k) Z(k) = Z (k 1) + k n k u k n 1 + (k ) n + 9n n 64 k u n + 5n n Z(k) < k=1 n(n + 1). n(n + 1) <. (-1). n 64 (-14). n < 64, n = 1 n = 3 (-14) : Z(n) = φ(n), (-) Z(n) + φ(n) = n (-3), φ(n) Euler., n p. n = p p 1(mod 4), n.,. n = 1. : Z(n) φ(n) ( ): (a) n > 1, n = p α 1 1 pα pα r r, p 1 < p < < p r, α i 1, i = 1,,, r n, n (-), α r = 1, (i) n = α p, n (-) p 1(mod 4), α 1 1(mod p); 94

102 Smarandache (ii) (-) p α 1 1 p (p 1, p ). (b) (i) n, (-3) n = α p, p 3(mod 4), α 1 1(mod p). (ii) n, n = p α 1 1 pα pα r r, p 1 < p < < p r, α i 1, i = 1,,, r n, n (-3), α r = 1, (-3)..3.0: S(Z(n)) = Z(S(n)).. Smarandache, Smarandache Smarandache., :.3.1: Z(n)..3.: Z(n) Smarandache..3.3: Z(n)., Smarandache,. 95

103 Smarandache Kenichiro Kashihara Kenichiro Kashihara Smarandache., Kenichiro Kashihara. : : C n C = lim ( 1 ln n). n k k=1 3.1: Smarandache s n ( ) n C s = lim ( 1 ln s n ). n s k k=1 3.:, p n n. n lim ( 1 ln p n ) n p k k=1 3. Smarandache 3.3: Smarandache,? : Kenichiro Kashihara..,,. 96

104 Kenichiro Kashihara 3.3,. 3.4: Smarandache s n,. [s 1, s, s 3, ] = s s + 1 (s 3 + ) : Smarandache,.,. 3.4 Dirichlet Dirichlet : 3.1: {a n }, a n = np + q, p q. {a n }. 3.1, Kenichiro Kashihara : 3.1: {b n } b n = p n +q, (p+q+1). {b n }. :,., (p + q + 1). Kenichiro Kashihara : {b n }

105 Smarandache 3.5 Smarandache Dirichlet 3.: {a n }, Dirichlet : s C. n N a n n s, 3.3: Smarandache Dirichlet Dirichlet, {a n } Smarandache. 3.5: Smarandache Dirichlet. 3.6 Kenichiro Kashihara, : 1) 3.4: {p n }, {x n } p x n n 1 0(mod p n+1 ) x n. :, 4, 6, 10, 1, 4, 9,, 7, 10, 4, 10, 7, 46, 13, 9, 60, 66, 70,. 3.: {x n },. 3.3: {x n },. 3.4: 8 {x n }. ) 3.5: {s n }, s n = n. {y n } s y n n 1 0(mod s n+1 ). : 98 1, 3,, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15,.

106 Kenichiro Kashihara, Kenichiro Kashihara {y n }, : 3.:, { k, n = k 1, y n = k + 1, n = k :, n y 1 = (n 1)(n y + n y n + 1) 0(mod (n + 1) ). (n 1, n + 1) = 1, (n y + n y n + 1) 0(mod (n + 1)), y = n + 1. (n 1, n + 1) =, (n y + n y n + 1) 0 y = n ) ( mod n + 1 ), 3.6: {c n }, c n = n 3. {z n } c z n n 1 0(mod z n+1 ) z n. : 1, 6, 16, 50, 6, 98, 64, 54, 50, 4, 1,., Kenichiro Kashihara : 3.5: {z n },. 3.6: {z n }. : [51] {z n }, z n. (3.5) (3.6), : 99

107 Smarandache 3.3: n, {z n }, : (a). x n = (n + 1), n 1 (mod 6) n 3 (mod 6), (b). x n = (n + 1), n 0 (mod 6) n 4 (mod 6), (c). x n = 1 3 (n + 1), n 5 (mod 6), (d). x n = 3 (n + 1), n (mod 6). x 1, x n (n > 1). (a) {x n }. Kenichiro Kashihara! : 3.5. n > 1, y (3-1) n 3y 1(mod (n + 1) 3 ). (3-1) n 3y 1 = (n + 1 1) 3y ( 1) 3y 1 ( 1) y 1 0 (mod (n + 1)). y! n > 1, x n x n, (3-1) y : n 3y 1 = (n + 1 1) 3y 1 3y(3y 1) (n + 1) 3y(n + 1) 0 (mod (n + 1) 3 ). (3-) 3y(n + 1) 0 (mod (n + 1) ). (3-3) : (3, n+1) = 1, (3-3) y = k(n+1), k. y = k(n + 1) (3-) 3k(3k(n + 1) 1) (n + 1) 3 3k(n + 1) 0 (mod (n + 1) 3 ). (3-4) y, n + 1, (3-4) k = n + 1. n+1, (3, n+1) = 1, n = 6t+1 n = 6t+3, t 100

108 Kenichiro Kashihara. n 6t + 1 6t + 3 ( t ), n 3x n 1 (mod (n + 1) 3 ) x n x n = (n + 1). n + 1, y (3-4) k = (n + 1). (6, n + 1) = 1, n = 6t n = 6t + 4, t. n 6t 6t + 4 ( t ), n 3x n 1 (mod (n + 1) 3 ) x n x n = (n + 1). (3, n+1) > 1, 3 n+1, (3-3) y = 1 3 k(n+1), k. y = 1 k(n + 1) (3-) 3 k(k(n + 1) 1) (n + 1) 3 k(n + 1) 0 (mod (n + 1) 3 ). (3-5) n + 1, (3-5) k = n n + 1, n = 6t + 5, n 6t + 5 ( t ), n 3x n 1 (mod (n + 1) 3 ) x n x n = 1 3 (n + 1). (, n + 1) = 1, y, (3-5) k = (n + 1). 3 n + 1, (, n + 1) = 1, n = 6t +, n 6t + ( t ), n 3x n 1 (mod (n + 1) 3 ) x n x n = 3 (n + 1). n 0, 1,, 3, 4, 5 (mod 6),! 3.7 Smarandache 3.6: Smarandache, : s n+1 s n < n i= s i. (3-6) 101

109 Smarandache {p n } {s n }, Kenichiro Kashihara (3-6). n (1 1 ) p i p i, i=1 p n+1 p n < n i= p i. {p n }. Kenichiro Kashihara,. 3.8 Smarandache 3.7: Smarandache {s n }, : n ρ R. lim n ( i=1 s i = ρ, n s i ) i=1 Kenichiro Kashihara, {p n } {s n }, ρ ρ : {a n }, {a n } {b n } : {b n } a b n+1 a n+ (mod a n ) b. 10

110 Kenichiro Kashihara 3.8: {a n s n }, {b n }. {a n }., : : x n = (p n, p n+1, p n+ ) p x n+1 p n+ (mod p n ) (, 3, 5) = 1, (3, 5, 7) =, (5, 7, 11) = 4, (11, 13, 17) = 9, (13, 17, 19) = 9, (19, 3, 9) = 16, (3, 9, 31) = 4, (9, 31, 37) = 3, (59, 61, 67) = 3, (71, 73, 79) = 3, (71, 73, 79) = 3, (79, 83, 89) = 18,., {x n }.,., {x n } : 11 x 13(mod 7) 4 x 6(mod 7) 3 4x 3 3 (mod 7) 4x 3(mod) 1,, 4,, 16, 4, 3,,,,,,, 3,,, 3,, 18, 81,,, 70,, : n N, p p n, n., n = 4, p = 5, n = 6, p = 11, 5 = ; 103

111 Smarandache 11 = : Lagrange : n,. ( p ) (n 4). Smarandache. Kenichiro Kashihara,,. 3.9: (n, p) p = x 1 + x + + x n, x 1 < x < < x n, n N, p., : U = {1} { }, m : m = i N a b j i, a i, b j U a i., b j, : 10 = 1 + 3, 31 = + 3 3, 59 = =

112 Kenichiro Kashihara,. 3.11: 3.10 : b j, Kenichiro Kashihara,. 3.1:. 3.1 p- p : q, N(q) N q, : q N, r, 0 < r < 1, N(q) r = lim N N. Kenichiro Kashihara, Diophantine Diophantine,. 3.14: x y + y z + z x = 0 (3-7). 105

113 Smarandache : (3-7), : (3-7), (x, y, z) = (, 1, 1), (1,, 1), (1, 1, ). (3-7), n x x 1 + xx xx n n 1 + xx 1 n = 0,! 3.15: (a b b a ) a+b = c. (a b b c c a ) 3 a+b+c = d : [5], : 3.4: a, b c (a b b a ) a+b = c (3-8) a = b = s, c = s, s. : a, b c, Diophantine, (a b b a ) a+b = c. (3-9) (a, b) = d a b, a = a 1 d, b = b 1 d, (a 1, b 1 ) = 1. (3-9), a b 1 a 1 +b 1 1 b a 1 a 1 +b 1 1 d = c. (3-10) a (3-10), (a 1, b 1 ) = 1, c, d, a 1 +b 1 1, b1. b 1 a 1 a 1 +b 1 106

114 Kenichiro Kashihara a 1 = b 1, (3-10) a 1 b 1 d = a = c. a = b, c = a Diophantine (3-8). a a 1 b 1, a 1 > b 1. a 1 +b 1 1., a 1 p, p α a 1, p α+1 a a 1. a 1 +b 1 1, p αb1 a 1 +b 1, αb 1 a 1 +b 1. (a 1, b 1 ) = 1, (a 1 + b 1, b 1 ) = 1 αb 1 a 1 +b 1, p α a 1, (3-11) b 1 a 1 + b 1 α. (3-11) α a 1 + b 1 > a 1 = p α a, (3-1) a. p, α, p α a p α α α, (3-1). a, b c Diophantine (3-8) a = b, c = a : 3 (a b b c c a ) a+b+c = d, (3-13) (a, b, c) = t, a = a 1 t, b = b 1 t, c = c 1 t (a 1, b 1 ) = (a 1, c 1 ) = (c 1, b 1 ) = 1, a, b, c, d ; (a, b, c, d) = (t, t, t, t 3 ), (t, t, 4t, t 3 ), (4t, t, t, t 3 ) (t, 4t, t, t 3 ), a, b, c, d, t. : a, b, c, d Diophantine (3-13). (a, b, c) = t, a = a 1 t, b = b 1 t, c = c 1 t (a 1, b 1, c 1 ) = 1, (3-13) 3b 1 3c 1 a 1 +b 1 +c 1 3a 1 a 1 +b 1 +c 1 a a 1 +b 1 +c 1 1 b1 c1 t 3 = d. (3-14) (3-14) a 1 = b 1 = c 1, (a 1, b 1, c 1 ) = 1, a 1 = b 1 = c 1 = 1, d = t 3. a = b = c, d = a 3 Diophantine (3-13). a 1 = b 1 < c 1, (3-14) b 1 3(a 1 +c 1 ) a a 1 +c 1 3a 1 a 1 +c 1 1 c1 t 3 = d. (3-15) 107

115 Smarandache 3(a 1 +c 1 ) 3a 1 a (a 1, c 1 ) = 1, a 1 +c 1 a 1 c 1 +c 1 1. c 1 p, p α c 1, p α+1 c 1, 3a 1α a 1 +c 1 (a 1 + c 1 ) 3a 1 α. (a 1 + c 1, c 1 ) = 1, (a 1 + c 1 ) 3α 3α = k(a 1 + c 1 ). 3α a 1 + c 1 = a 1 + p α c a 1 + p α 3α a 1 p α. a 1 = 1, p = = α, a = b = t, c = 4t, d = t 3., t, a = b = t, c = 4t, d = t 3 (3-13). a 1 = b 1 > c 1, a 1 p, p β a 1, p β+1 a 1, a 3(a 1 +c 1 ) a 1 +c 1 1 (a 1 + c 1 ) 3(a 1 + c 1 )β. (a 1 + c 1, a 1 + c 1 ) = 1, (a 1 + c 1 ) 3β 3β = k(a 1 + c 1 ). 3β a 1 + c 1, 3β p β., β, 3β 1 < p β. a 1 > b 1 > c 1, (a 1, b 1 ) = (a 1, c 1 ) = (c 1, b 1 ) = 1, (3-14)., (a 1, b 1 ) = (a 1, c 1 ) = (c 1, b 1 ) = 1, 3b 1 3c 1 a 1 +b 1 +c 1 a a 1 +b 1 +c 1 1, b1, c 3a 1 a 1 +b 1 +c 1. (a 1, b 1 + c 1 ) = u, (b 1, a 1 + c 1 ) = v, (c 1, a 1 + b 1 ) = w. (a 1, b 1, c 1 ) = 1, (u, v) = (v, w) = (w, u) = 1. c 1 = 3a 1 a 1, b 1 +b 1 +c 1 1., c 1 3. u 3a 1 a a1, c 1 +b 1 +c 1 1., c p, p α c 1,p α+1 c 1, 3a 1α a 1 +b 1 +c 1, 3a 1 α = t(a 1 + b 1 + c 1 ) 3a 1α u = t a1+b 1 +c 1 u. ( a 1 u ), a 1+b 1 +c 1 u ) = 1, a 1 u t. 3α = t a1 u (a 1 + b 1 + c 1 ) a 1 + b 1 + c 1 3c p α + 3.., v b 1 w a c 1, a 1 +b 1 +c 1 1 b. u > a 1, v > b 1, w > c b 1 3c 1 a 1 +b 1 +c 1 1

116 Kenichiro Kashihara a 1 < u a 1, u a 1 + b 1 + c 1, v a 1 + b 1 + c 1 w a 1 + 3a 1 a b 1 + c 1, c 1 +b 1 +c 1 1., p, p α c 1,p α+1 c 1, 3a 1 α a 1 +b 1 +c 1, 3a 1 α = t(a 1 + b 1 + c 1 ). 3α = t a1 u v w x v w c 1 p α., c 1 < w c 1 a, b 1 +b 1 +c 1 1. w > c 1, u > a 1, u a 1, w c 1, a 1 = u, c 1 = w. u a 1 + b 1 + c 1, v a 1 + b 1 + c 1 (u, w) = 1, u w a 1 + b 1 + c 1. a 1 + b 1 + c 1 u w = a 1 c 1 3a 1 a 1 + b 1 + c 1.., a 1 > b 1 > c 1 (a 1, b 1 ) = (a 1, c 1 ) = (c 1, b 1 ) = 1, (3-14)., (a, b, c) = t, a = a 1 t, b = b 1 t, c = c 1 t (a 1, b 1 ) = (a 1, c 1 ) = (c 1, b 1 ) = 1, a, b, c (3-13). (a, b, c, d) = (t, t, t, t 3 ), (t, t, 4t, t 3 ), (4t, t, t, t 3 ) (t, 4t, t, t 3 ), a, b, c, d (3-13) (3-13). 3.16:, 3 n N. 3c 1 y n = nx n + 1 (3-16) : Kenichiro Kashihara. n = (3-15)., n,. 3.8: n 3, (3-15). 3.17: (1 + ) n (1 + 3) n 109

117 Smarandache, x x. :. (1 + ) n, (1 ) n ).,,, 1).,.,.,,. 110

118 Smarandache Smarandache 4.1, Smarandache,. 4. Goldbach, Goldbach. 4.1: n, n = p + q r, p, q, r. : p = p + q r. : 1 = = = = =, 3 = = = =, 5 = =, 7 = =. 4.1: n 9, Goldbach 4.:, 4.3: 4.: : n, n = p q r, p, q, r. 1 = = = =, 3 = = =, 111

119 Smarandache 5 = =, 7 = =. 4.4: Goldbach 4.5:, 4.6: 4.3: n, n = p + q + r + t u, p, q, r, t, u, t u. : 1 = = =, 3 = =, 5 = =, 7 = =. 4.7:, 4.8: 4.4: n, n = p + q + r t u, p, q, r, t, u, t, u p, q, r. : 1 = = =, 4.9: = =, 5 = =, 7 = =. 4.5: n, n = p + q r t u, p, q, r, t, u, r, t, u p, q. 11

120 Smarandache : 1 = =, 4.10: = =, 5 = =, 7 = =. 4.6: n, n = p q r t u, p, q, r, t, u, q, r, t, u p. : 1 = =, 4.11: = =, 5 = =, 7 = =. 4.7: k 3, 1 < s < k. (a) k, n, n = p 1 + p + + p k s q 1 q q s, p i q j, i = 1,,, k s, j = 1,,, s. 4.1: k, 4.13: k (b) k, n, n = p 1 + p + + p k s q 1 q q s, p i q j, i = 1,,, k s, j = 1,,, s. 4.14: : : s 1, s, s 3,, s n,, s 1, s 1 s, s 1 s s 3,. 113

121 Smarandache H.Ibstedt, S S. 4.4 f, R K. 4.15: n : n = R(f(n 1 ), f(n ),, f(n k )). 4.16:, k n 1, n,, n k n 1 + n + + n k = n. n :. :, 3, 5, 7, 3, 37, 53, 73, 3, 7, 33, 57, 77,. 4.17: Sylvester Smith : x y +y x, gcd(x, y) = 1, gcd x y. F.Luca : 4.1: a, b, c, ax y + by x = cz n 114

122 Smarandache (x, y, z, n), x, y, n gcd(x, y) = 1, gcd x y : S, S, f S S, : a 1 = f(s), s S ; a = f(a 1 ) = f(f(s)); a 3 = f(a ) = f(f(a 1 )) = f(f(f(s))),. 4.19: S f,. 4.8 Numberical Carpet 1 1 a 1 1 a b a 1 1 a b c b a 1 1 a b c d c b a 1 1 a b c d e d c b a 1 1 a b c d c b a 1 1 a b c b a 1 1 a b a 1 1 a ,., 1 a, a ; a. b, 115

123 Smarandache b ; a.., g. : : C(n, k) n k, n 0, k 0. C(n, 0) = 1; C(n, 1) = 4(n + 1); C(n, k) = 4(n + 1) k (4n + 1 4i), k n + 1. i=0, {C(n, k)} ( Scientia Magna ): n, p, 4n 4k + 9[ p 4n ] + 1 n k + p 1 mod, C(n, k) 0 mod p; n k p [ ] 3 n + 1 n + p 3 mod 4, C(n, k) 0 mod p : :, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 1,, 3, 9, 31, 33, 34, 35, 37, 38, 39, 41, 4, 43, 46, 51, 53, 55,. ( 0 1) kp, p, k. 116

124 Smarandache 4.0:., : 4.1: ( 0 1) kp, p, k. :, 3, 4, 5, 6, 7, 9, 10, 11, 1, 13, 14, 15, 17, 18, 19, 0, 1,, 3, 5, 6, 8, 9, 30, 31, 33, 34,. 4.1: : m ( 0 1) kp m, p, k, m., m =,. 4.: : n,,. : 7, 13, 19, 3, 5, 31, 33, 37, 43, 47, 9, 53, 55,. 4.3: : ( 0 1) k (k )(,4, 8, 16, 3, 64, ), 3 k (k ),, k(k ). :, 3, 5, 7, 10, 11, 1, 13, 14, 15, 17, 18, 19, 0,. 4.4:

125 Smarandache 4.10 Syllabic Puzzle 4.16: Syllabic Puzzle : 1, 1, 1, 1, 1, 1,,,, 1, 3, 1, 3, 3, 3, 3, 4, 3, 4,,. n n. 4.5: Code Puzzle 4.17: : A, B, C, D, E, F, G, H, I, J, K, L, M, N, O,, 01, 0, 03, 04, 05, 06, 07, 08, 09, 10, 11, 1, 13, 14, 15,. n n, Code Puzzle. : 1 = ONE = : 51405, 0315, , , ,. 4.6:. 4.7: : a(n) = k, k n, k+1 n. : 0, 1, 0,, 0, 1, 0, 3, 0, 1, 0,, 0, 1, 0, 4,. 4.8:

126 Smarandache 4.19: 3 b(n) = k, 3 k n, 3 k+1 n. 4.9: :. p c(n) = k, p k n, p k+1 n, p 4.30: : n,,. : 1,, 6, 10, 0, 4, 4, 60, 100, 10, 10, 00,. :,. 4.: n,,. : 10, 0, 4, 60, 100, 10, 00, 01, 04, 07, 10, 40,. 4.3: n,,. : 10, 0, 4, 60, 100, 10, 00, 01, 04, 07, 10, 40,. 4.31:. 4.3: 119

127 Smarandache : n, n, n. : 1, 10, 100, 1,, 10, 0, 100, 00, 1, 3, 100,. : n, = 1, 1 4.5: n, n n, n. : 10, 100, 10, 0, 100, 00, 10, 30, 100, 300, 10, 0,. : = 1, 1 4.6: n, n, n. : 10, 100, 10, 0, 100, 00, 10, 30, 100, 300, 10, 0,. 4.33: : n,,. 10

128 Smarandache : 0,, 4, 6, 8, 10, 1, 14, 16, 18, 0, 1,. :,. 4.8: n,,. : 1, 3, 5, 7, 9, 41, 43, 45, 47, 49, 61, 63,. 4.9: n,,. : 0, 1,, 3, 4, 5, 6, 7, 8, 9, 40, 41,. 4.34: : n,,. : 1, 3, 5, 7, 9, 10, 11, 1, 13, 14, 15, 6,. 4.31: n,,. : 10, 1, 14, 16, 18, 30, 3, 34, 36, 38,. 4.3: n,,. 11

129 Smarandache : 10, 11, 1, 13, 14, 15, 16, 17, 18, 19,. 4.35: : n, k, n, n. : n n,. 4.34: n, k, n, n, n. 4.35: n, k, n, n. 4.36: triangular number 4.36: n, triangular number, triangular number. : triangular number : n(n + 1). 4.37: Smarandache-Kurepa 4.37: p, SK(P ) p!sk(p ),!SK(P ) = 0! + 1! +! + + (p 1)!. 1 SK(P ) :

130 Smarandache n SK(P ) n SK(P ) n SK(P ) :. 4.0 Smarandache-Wagstaff 4.38: p, SW (P ) p W (SK(P )), W (P ) = 0! + 1! +! + + (p)!. SW (P ) : n SW (P ) n SW (P ) n SW (P ) :. 4.1 n Smarandache. 4.39: n, SK(n) n SK(n) k 1 n 1, SK(n) : n SK(n) n SK(n) n SK(n)

131 Smarandache 4.40:. 4. Smarandache Near-To-Primordial 4.40: n, SNT P (n) n p 1, p, p n 8, SNT P (n) : n SNT P (n) n SNT P (n) :. 14

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135 On The Smarandache Notions And Related Problems Wang Yu Department of Mathematics Northwest University Xi an, Shaanxi, P. R. China Su Juanli Department of Pulic Teaching Yangling Vocational and Technical College Yangling, Shaanxi, P. R. China Zhang Jin Department of Mathematics Xi an Normal School Xi an, Shaanxi, P.R.China High American Press 008

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