Introduction to Lucas Sequences
|
|
- Kevin Lindsey
- 5 years ago
- Views:
Transcription
1 A talk given at Liaoning Normal Univ. (Dec. 14, 017) Introduction to Lucas Sequences Zhi-Wei Sun Nanjing University Nanjing 10093, P. R. China zwsun Dec. 14, 017
2 Abstract Let A and B be integers. The Lucas sequences u n = u n (A, B) (n = 0, 1,,...) and v n = v 0 (A, B) (n = 0, 1,,...) are defined by and u 0 = 0, u 1 = 1, u n+1 = Au n Bu n 1 (n = 1,, 3,...) v 0 =, v 1 = A, v n+1 = Av n Bv n 1 (n = 1,, 3,...). They are natural extensions of the Fibonacci numbers and the Lucas numbers. Lucas sequences play important roles in number theory and combinatorics. In this talk we introduce various properties of Lucas sequences and their applications. In particular, we will mention their elegant application to Hilbert s Tenth Problem on Diophantine equations. / 41
3 Fibonacci numbers In a book in 10, Fibonacci considered the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year? Suppose that in the n-th month there are totally F n pairs of rabbits. Then Note that F 0 = 0, F 1 = 1, F n+1 = F n + F n 1 (n = 1,, 3,...). F = 1, F 3 =, F 4 = 3, F 5 = 5, F 6 = 8, F 7 = 13, F 8 = 1, F 9 = 34, F 10 = 55, F 11 = 89, F 1 = / 41
4 A combinatorial interpretation of Fibonacci numbers Let f (n) denote the number of binary sequences not containing two consecutive zeroes. Clearly, f (1) = (both 0 and 1 meet the purpose), and f () = 3 (01, 10, 11 meet the purpose). Now let a 1,..., a n {0, 1}. Clearly a 1..., a n 1 meets the purpose if and only if the sequence a 1..., a n meets the purpose. Also, the sequence a 1..., a n 0 meets the purpose if and only if a n = 1 and the sequence a 1... a n 1 meets the purpose. Therefore f (n + 1) = f (n) + f (n 1). Since f (1) = F 3, f () = F 4 and also f (n + 1) = f (n) + f (n 1) for n = 1,, 3,..., we see that f (n) = F n+. 4 / 41
5 Edouard Lucas The name Fibonacci numbers or the Fibonacci sequence was first used by the French mathematician E. Lucas ( ). Lucas has several fundamental contributions to number theory. Lucas Congruence. Let p be a prime, and let a = k i=0 a ip i and b = k i=0 b ip i with a i, b i {0,..., p 1}. Then ( ) a k ( ) ai (mod p). b i=0 b i Lucas died in unusual circumstances. At the banquet of an annual congress, a waiter dropped some crockery and a piece of broken plate cut Lucas on the cheek. He died a few days later of a severe skin inflammation probably caused by septicemia. He was only / 41
6 Lucas sequences Let A and B be given numbers. For n N = {0, 1,,...} we define u n = u n (A, B) and v n = v n (A, B) as follows: u 0 = 0, u 1 = 1, and u n+1 = Au n Bu n 1 (n = 1,, 3,...); v 0 =, v 1 = A, and v n+1 = Av n Bv n 1 (n = 1,, 3,...). The sequence {u n } n N and its companion {v n } n N are called Lucas sequences. In 1876 E. Lucas introduced such sequences in the case A, B Z, and studied them systematically. Lucas sequences play important roles in number theory. 6 / 41
7 Special cases By induction on n N we find that u n (, 1) = n, v n (, 1) = ; u n (3, ) = n 1, v n (3, ) = n Those F n = u n (1, 1) and L n = v n (1, 1) are called Fibonacci numbers and Lucas numbers respectively. The Pell sequence {P n } n N is given by P n = u n (, 1), so P 0 = 0, P 1 = 1, P n+1 = P n + P n 1 (n = 1,, 3,...). The companion of the Pell sequence is {Q n } n N where Q n = v n (, 1). We also let S n = u n (4, 1) and T n = v n (4, 1); thus S 0 = 0, S 1 = 1, and S n+1 = 4S n S n 1 (n = 1,, 3,...); T 0 =, T 1 = 4, T n+1 = 4T n T n 1 (n = 1,, 3,...). 7 / 41
8 Binet Formulae The equation x = Ax B is called the characteristic equation of the Lucas sequences {u n (A, B)} n N and {v n (A, B)} n N. = A 4B is the discriminant, and the two roots are α = A + and β = A. Property 1 (Binet, 1843) For any n = 0, 1,,... we have u n = α k β n 1 k and v n = α n + β n. 0 k<n As A = α + β and B = αβ, we can prove the formulae by induction on n. If = 0, then α = β = A/, hence u n = ( ) A n 1 ( ) A n α n 1 = n and v n = α n =. 0 k<n 8 / 41
9 Explicit Formulae We can express u n and v n in terms of n, A and. In fact, ( un =(α β)u n = α n β n A + ) n ( = and thus = n 1 n k=0 k n 1 u n = ( ) n A n k (k 1)/ k (n 1)/ k=0 ( ) n A n 1 k k k + 1 (which also holds when = 0). Similarly, ( v n =α n + β n A + ) n ( A ) n = + = 1 n 1 n/ k=0 ( ) n A n k k. k A ) n 9 / 41
10 Special cases For the Fibonacci sequence {F n } n N and its companion {L n } n N, = 1 4( 1) = 5 and so 5Fn = ( 1 + ) n 5 ( 1 ) n ( 5, L n = 1 + ) n 5 +( For the Pell sequence {P n } n N and its companion {Q n } n N, = 4( 1) = 8 and so P n = (1 + ) n (1 ) n, Q n = (1 + ) n + (1 ) n. 1 ) n 5. For the sequence {S n } n N and its companion {T n } n N, = = 1 and so S n = 1 ( ( + 3) n ( 3) n), T n = (+ 3) n +( 3) n / 41
11 On ((A + A 4B)/) n Let A, B Z with = A 4B. For n N, as ( A + ) n ( A ) n un (A, B) =, ( A + ) n ( A ) n v n (A, B) = +, we have In particular, ( A ± ) n = v n(a, B) ± u n (A, B). ( 1 ± ) n 5 = L n ± F n 5, (1 ± ) n = Q n ± P n, ( ± 3) n = T n ± S n / 41
12 Relations between u n and v n Property. For n N we have v n = u n+1 Au n, u n = v n+1 Av n, vn un = 4B n. This can be easily proved since A = α + β, u n = α n β n, v n = α n + β n, where = A 4B, α = A + and β = A. In particular, L n = F n+1 F n, 5F n = L n+1 L n, L n 5F n = 4( 1) n. 1 / 41
13 Neighbouring formulae Property 3 (Neighbouring formulae). For any n N we have un+1 Au n+1 u n + Bun = B n, vn+1 Av n+1 v n + Bvn = B n. In other words, if n Z + = {1,, 3,...} then un u n 1 u n+1 = B n 1 and vn v n 1 v n+1 = B n 1. Proof. By Property, 4B n = vn un = (u n+1 Au n ) un and 4 B n = vn ( u n ) = vn (v n+1 4Av n ). So the desired results follow. Example. Fn+1 F n+1 F n Fn = ( 1) n, L n+1 L n+1 L n L n = 5( 1) n / 41
14 Addition formulae Property 4 (Addition Formulae). For any m, n N we have u m+n = u mv n + u n v m and v m+n = v mv n + u m u n. Property 5 (Double Formulae). For each n N we have u n = u n v n, v n = v n B n = v n + u n = u n + B n, u n+1 = u n+1 Bu n, v n+1 = v n v n+1 AB n. 14 / 41
15 Multiplication Formulae Property 6 (Multiplication Formulae). For any k, n N we have u kn = u k u n (v k, B k ) and v kn = v n (v k, B k ). Proof. Let A = v k and B = B k. Then = v k 4Bk = u k, α = v k + u k, β = v k u k. Hence ( (α ) n + (β ) n v k + ) n ( u k v k ) n u k = + ( A + ) kn ( A ) kn = + = v kn. That u kn = u k u n (v k, B k ) can be proved similarly. 15 / 41
16 Expansions in terms of A and B Property 7 (Expansions in terms of A and B). For any n Z + we have (n 1)/ ( ) n 1 k u n = A n 1 k ( B) k, k k=0 n/ n v n = n k k=0 Proof. Observe that u m x m 1 = m=1 = m 1 m=1 k=0 m 1 ( n k k α k β m 1 k x m 1 ) A n k ( B) k. (αx) k (βx) m 1 k = m=1 k=0 (αx) k k=0 j=0 (βx) j = 1 1 αx 1 1 βx = 1 1 Ax + Bx. 16 / 41
17 Expansions in terms of A and B (continued) Let [x m ]f (x) denote the coefficient of x m in the power series for f (x). Then u n 1 =[x n 1 ] u m x m 1 = [x n 1 1 ] 1 Ax + Bx m=1 =[x n 1 1 (x(a Bx))n ] 1 x(a Bx) n 1 = [x n 1 ] n 1 =[x n 1 ] x n 1 k (A Bx) n 1 k = = k=0 (n 1)/ k=0 (n 1)/ k=0 ( n 1 k k ( n 1 k k k=0 ) ( B) k A n 1 k k ) A n 1 k ( B) k. x k (A Bx) k 17 / 41
18 Expansions in terms of A and B (continued) v n =u n+1 Au n n/ ( ) n k = A n k ( B) k k = k=0 A n/ k=0 (n 1)/ ( k=0 n/ n = n k k=0 ( n k k ( n 1 k k ) ( n k k ) A n 1 k ( B) k ( n 1 k k ) A n k ( B) k. Corollary. For any n Z + we have F n = ( ) n 1 k and L n = k k N 0 k n/ )) A n k ( B) k n n k ( n k k ). 18 / 41
19 On linear index Property 8 (On linear index) (Z.-W. Sun [Sci. China Ser. A 35(199)]). Suppose that w n+1 = Aw n Bw n 1 for n = 1,, 3,.... Then, for any k Z + and l, n N we have n ( ) n w kn+l = ( Bu k 1 ) n j u j j k w l+j, in particular w n+l = j=0 j=0 n j=0 ( ) n ( B) n j A j w l+j. j Corollary. For any n N we have n ( ) n F n = F j and F n+1 = j n j=0 ( ) n F j+1. j 19 / 41
20 On linear index (continued) We prove the identity by induction on n. The case n = 0 is trivial. Now assume that the identity for n is valid. Then n ( ) n w k(n+1)+l =w kn+(k+l) = ( Bu k 1 ) n j u j j k w k+l+j j=0 n ( ) n = ( Bu k 1 ) n j u j j k (u kw l+j+1 Bu k 1 w l+j ) j=0 =u n+1 k + + j=0 n j=1 n j=1 ( ) n ( Bu k 1 ) n+1 j u j j 1 k w l+j ( ) n ( Bu k 1 ) n+1 j u j j k w l+j + ( Bu k 1 ) n+1 w l n+1 ( ) n + 1 = ( Bu k 1 ) n+1 j u j j k w l+j. 0 / 41
21 Lucas Theorem Property 9 (E. Lucas) Let A, B Z with (A, B) = 1. Then (u m, u n ) = u (m,n) for all m, n N. Proof. By induction, u n+1 A n (mod B). As (A, B) = 1, we have (u n+1, B) = 1. Since u n+1 Au n+1u n + Bu n = B n by Property 3, (u n, u n+1 ) B n and hence (u n, u n+1 ) (u n+1, B n ) = 1. By Property 8, for any n Z + and q, r N we have u nq+r = q j=0 ( ) q ( Bu n 1 ) q j u j j nu r+j ( Bu n 1 ) q u r = (u n+1 Au n ) q u r u q n+1 u r (mod u n ) and hence (u nq+r, u n ) = (u n, u r ). 1 / 41
22 Lucas Theorem (continued) Clearly, (u m, u 0 ) = (u m, 0) = u m = u (m,0). Now let r 0 = m and r 1 = n Z +. Write r i 1 = r i q i + r i+1 for i = 1,..., k with r 1 > r >... > r k > r k+1 = 0. For each i = 1,..., k, Therefore, (u ri 1, u ri ) = (u qi r i +r i+1, u ri ) = (u ri, u ri+1 ). (u m, u n ) =(u r0, u r1 ) = (u r1, u r ) =... =(u rk, u rk+1 ) = (u (m,n), u 0 ) = u (m,n). Remark. For m, n Z +, we can prove that { v (v m, v n ) = (m,n) if ord (m) = ord (n), (, v (m,n) ) otherwise. / 41
23 On positivity of u n and v n Property 10 (Z.-W. Sun [PhD thesis, 199]). u n 0 for all n N v n 0 for all n N A 0 and = A 4B 0. Proof. If A 0 and 0, then u n 0 and v n 0 by the explicit formulae in terms of A and. Suppose that u n 0 for all n N. Then A = u 0. If < 0, then u n+1 u nu n+ = B n > 0 and the decreasing sequence (u n+1 /u n ) n 1 has a real number limit θ. Since u n+ u n+1 = Au n+1 Bu n u n+1 = A B u n+1 /u n (n = 1,, 3,...), we see that θ Aθ + B = 0 and thus 0. Similarly, if v n 0 for all n N then A = v 0 0 and 0. 3 / 41
24 Periodicity of u n and v n modulo m Property 11 (Periodicity of u n and v n modulo m). Let m Z + with (B, m) = 1. Then there is a positive integer λ such that and also u n+λ u n (mod m) for all n N v n+λ v n (mod m) for all n N. Proof. Consider the m + 1 ordered pairs u i, u i+1 (i = 0,..., m ). By the Pigeon-hole Principle, two of them are congruent modulo m. Choose the least λ m such that for some 0 j < λ. Since u λ, u λ+1 u j, u j+1 (mod m) Bu j 1 = u j+1 Au j u λ+1 Au λ = Bu λ 1 (mod m) and (B, m) = 1, we get u j 1 u λ 1 (mod m). Continuing this process, we finally obtain u λ j, u λ j+1 u 0, u 1 (mod m). 4 / 41
25 Periodicity of u n and v n modulo m (continued) By the choice of λ, we must have j = 0 and thus u λ, u λ+1 u 0, u 1 (mod m). It follows that u n+λ u n (mod m) for all n = 0, 1,,.... For any n N, we also have v n+λ = u n+λ+1 Au n+λ u n+1 Au n = v n (mod m). This completes the proof. 5 / 41
26 Legendre symbols Let p be an odd prime and a Z. The Legendre symbol ( a p ) is given by ( ) a 0 if p a, = 1 if p a and x p a (mod p) for some x Z, 1 if p a and x a (mod p) for no x Z. It is well known that ( ab p ) = ( a p )( b p ) for any a, b Z. Also, ( ) { 1 = ( 1) (p 1)/ 1 if p 1 (mod 4), = p 1 if p 1 (mod 4); ( ) { = ( 1) (p 1)/8 1 if p ±1 (mod 8), = p 1 if p ±3 (mod 8). The Law of Quadratic Reciprocity: If p and q are distinct odd primes, then ( ) ( ) p q = ( 1) p 1 q 1. q p 6 / 41
27 Lucas sequences modulo primes Property 1. Let A, B Z with = A 4B, and let p be an odd prime. (i) u p ( p ) (mod p) and v p A (mod p). (ii) If p B, then p u p ( p ). Proof. Note that p ( ) p j for all j = 1,..., p 1. Thus (p 1)/ ( ) p u p p 1 u p = A p 1 k k k + 1 k=0 ( ) (p 1)/ (mod p) p and v p p 1 v p = (p 1)/ k=0 ( ) p A p k k k A p A (mod p). 7 / 41
28 Lucas sequences modulo primes Now we prove p u p ( p If p, then u p ( p ) = u p ( p If ( p ) = 1, then u p+1 = Au p + v p A ) under the condition p B. ) = 0 (mod p). and hence u p ( p ) = u p+1 0 (mod p). In the case ( p ) = 1, we have ( ) + A = 0 (mod p) p Bu p 1 = Au p u p+1 = Au p v p A (u p 1) 0 (mod p) and hence u p ( p ) = u p 1 0 (mod p) since p B. 8 / 41
29 Wall-Sun-Sun primes In 1960 D. D. Wall [Amer. Math. Monthly] investigated Fibonacci numbers modulo m. For d Z + let n(d) be the least n Z + such that d F n. Wall asked whether n(p) n(p ) for any odd prime p. Let p be an odd prime. Then F p ( p 5 ) = F p ( 5 ) 0 (mod p). p Z.-H. Sun and Z.-W. Sun [Acta Arith. 60(199)] proved that n(p) n(p ) p F p ( p 5 ) = x p + y p = z p for no x, y, z Z with p xyz. An odd prime p with p F p ( p 5 ) is called a Wall-Sun-Sun prime. There are no known Wall-Sun-Sun primes though heuristic arguments suggest that there should be infinitely many Wall-Sun-Sun primes. 9 / 41
30 Chebyshev polynomials The first kind of Chebyshev polynomials T n (x) (n N) and the second kind of Chebyshev polynomials T n (x) (n N) are given by Clearly, As cos nθ = T n (cos θ) and sin((n + 1)θ) = sin θ U n (cos θ). T 0 (x) = 1, T 1 (x) = x, T (x) = x 1, U 0 (x) = 1, U 1 (x) = x, U (x) = 4x 1. cos(nθ+θ) = cos θ cos nθ sin θ sin nθ = cos θ cos nθ cos(nθ θ) and sin(nθ +θ) = cos θ sin nθ +sin θ cos nθ = cos θ sin nθ sin(nθ θ), we have T n+1 (x) = xt n (x) T n 1 (x) and U n+1 (x) = xu n (x) U n 1 (x). 30 / 41
31 Chebyshev polynomials (continued) As T 0 (x) =, T 1 (x) = x, T n+1 (x) = xt n (x) T n 1 (x) (n = 1,,...), and U 0 (x) = 1, U 1 (x) = x, U n+1 (x) = xu n (x) U n 1 (x) (n = 1,,...), we see that T n (x) = v n (x, 1) and U n (x) = u n+1 (x, 1). Thus, for any n Z +, by Property 7 we have T n (x) = 1 n/ ( ) n n k (x) n k ( 1) k n k k and U n (x) = k=0 n/ k=0 ( n k k ) (x) n k ( 1) k. 31 / 41
32 Pell s equation Let d Z + \. It is well-known that the Pell equation y dx = 1 has infinitely many integral solutions. (Note that x = 0 and y = ±1 are trivial solutions.) Moreover, {y + dx : x, y Z and y dx = 1} is a multiplicative cyclic group. For any integer A, the solutions of the Pell equation y (A 1)x = 1 (x, y N) are given by x = u n (A, 1) and y = v n (A, 1) with n N. J. Robinson and his followers wrote u n (A, 1) and v n (A, 1) as ψ n (A) and χ n (A) respectively. To unify Matiyasevich s use of F n = u n (3, 1) and Robinson s use of ψ n (A) = u n (A, 1), we deal with Lucas sequences (u n (A, 1)) n 0. 3 / 41
33 On u n (A, 1) with n Z We extend the sequences u n = u n (A, 1) and v n = v n (A, 1) to integer indices by letting and u 0 = 0, u 1 = 1, and u n 1 + u n+1 = Au n for all n Z, v 0 =, v 1 = A, and v n 1 + v n+1 = Av n for all n Z. It is easy to see that u n (A, 1) = u n (A, 1) = ( 1) n u n ( A, 1) and v n (A, 1) = v n (A, 1) = ( 1) n v n ( A, 1) for all n Z. Lemma. Let A, X Z. Then (A 4)X + 4 X = u m (A, 1) for some m Z. Remark. For n N and A, it is easy to show that (A 1) n u n+1 (A, 1) A n. 33 / 41
34 Hilbert s Tenth Problem In 1900, at the Paris conference of ICM, D. Hilbert presented 3 famous mathematical problems. He formulated his tenth problem as follows: Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers. In modern language, Hilbert s Tenth Problem (HTP) asked for an effective algorithm to test whether an arbitrary polynomial equation P(z 1,..., z n ) = 0 (with integer coefficients) has solutions over the ring Z of the integers. However, at that time the exact meaning of algorithm was not known. 34 / 41
35 Two key steps to solve HTP Based on the above theorem, M. Davis, H. Putnam and J. Robinson [Ann. of Math. 1961] successfully showed that any r.e. set is exponential Diophantine, that is, any r.e. set A has the exponential Diophantine representation a A x x n 0[P(a, x 1,..., x n, x 1,..., xn ) = 0], where P is a polynomial with integer coefficients. Recall that the Fibonacci sequence (F n ) n 0 defined by F 0 = 0, F 1 = 1, and F n+1 = F n + F n 1 (n = 1,, 3,...) increases exponentially. In 1970 Yu. Matiyasevich took the last step to show ingeniously that the relation y = F x (with x, y N) is Diophantine! It follows that the exponential relation a = b c (with a, b, c N, b > 1 and c > 0) is Diophantine, i.e. there exists a polynomial P(a, b, c, x 1,..., x n ) with integer coefficients such that a = b c x x n 0[P(a, b, c, x 1,..., x n ) = 0]. 35 / 41
36 A key lemma The following lemma is an extension of a lemma of Yu. Matiyasevich (1970). Lemma (Sun [Sci. China Ser. A 35(199)]). Let A, B Z with (A, B) = 1, and let k, m N. (i) ku k m = u k u m. (ii) Suppose that A 0, A 4B, and A 1 or (k )B 0. Then u k u m ku k m. 36 / 41
37 Diophantine representation of C = u B (A, 1) with unknowns arbitrarily large Matiyasevi c and Robinson (1975) showed that for A > 1 and B, C > 0 there is a Diophantine representation of C = u B (A, 1) only involving three natural number variables. Lemma (Sun [Sci. China Ser. A 35(199)]). Let A, B, C Z with A > 1 and B 0. Then C = u B (A, 1) C B x > 0 y > 0(DFI ) x, y, z 0[DFI (C B + 1) = (z DFI (C B + 1)) ], where D = (A 4)C + 4, E = C Dx, F = 4(A 4)E + 1, G = 1 + CDF (A + )(A ) E, H = C + BF + (y 1)CF, I = (G 1)H + 1. Moreover, if C = u B (A, 1) with B > 0, then for any Z Z + there are integers x Z and y Z with DFI. 37 / 41
38 Diophantine representation of C = u B (A, 1) with integer unknowns Clearly C B x 0(C = B + x). However, if we use integer variables, we need three variables: C B x y z[c = B + x + y + z + z]. Thus, to save the number of integer variables involved, we should try to avoid inequalities. Note that u B (A, 1) u B (, 1) = B (mod A ). Lemma (Sun [Sci. China Ser. A 35(199)]). Let A, B, C Z with 1 < B < A / 1. Then C = u B (A, 1) (A C B) x 0 y(dfi ), where D, F, I are defined as before. 38 / 41
39 Diophantine representation of W = V B with integer unknowns J. Robinson showed that W = V B (with V > 1 and B, W > 0) if and only if there is an integer A > max{v 3B, W B } such that (V 1)W u B (A, 1) V (W 1) (mod AV V 1). Lemma (Sun [Sci. China Ser. A 35(199)]). Let B, V, W be integers with B > 0 and V > 1. Then W = V B if and only if there are A, C Z for which A max{v 4B, W 4 }, C = u B (A, 1) and (V 1)WC V (W 1) (mod AV V 1). Remark. A, V and W in this lemma are not necessarily positive, they might be negative. 39 / 41
40 Applications to Hilbert s Tenth Problem Matiyasevich (1975). There is no algorithm to determine for any P(x 1,..., x 9 ) Z[x 1,..., x 9 ] whether P(x 1,..., x 9 ) = 0 has solutions with x 1,..., x 9 N. Z.-W. Sun (199). There is no algorithm to determine for any P(z 1,..., z 11 ) Z[z 1,..., z 11 ] whether the equation has integral solutions. P(z 1,..., z 11 ) = 0 40 / 41
41 References For main sources of my work mentioned here, you may look at: 1. Z.-W. Sun, Reduction of unknowns in Diophantine representations, Sci. China Math. 35(199), Z.-W. Sun, Further results on Hilbert s tenth problem, arxiv: , Thank you! 41 / 41
A talk given at the City Univ. of Hong Kong on April 14, ON HILBERT S TENTH PROBLEM AND RELATED TOPICS
A talk given at the City Univ. of Hong Kong on April 14, 000. ON HILBERT S TENTH PROBLEM AND RELATED TOPICS Zhi-Wei Sun Department of Mathematics Nanjing University Nanjing 10093 People s Republic of China
More informationSome sophisticated congruences involving Fibonacci numbers
A tal given at the National Center for Theoretical Sciences (Hsinchu, Taiwan; July 20, 2011 and Shanghai Jiaotong University (Nov. 4, 2011 Some sohisticated congruences involving Fibonacci numbers Zhi-Wei
More informationSome New Diophantine Problems
A talk given at the Confer. on Diophantine Analysis and Related Topics (Wuhan, March 10-13, 2016) Some New Diophantine Problems Zhi-Wei Sun Nanjing University Nanjing 210093, P. R. China zwsun@nju.edu.cn
More informationHilbert s problems, Gödel, and the limits of computation
Hilbert s problems, Gödel, and the limits of computation Logan Axon Gonzaga University April 6, 2011 Hilbert at the ICM At the 1900 International Congress of Mathematicians in Paris, David Hilbert gave
More informationA talk given at University of Wisconsin at Madison (April 6, 2006). Zhi-Wei Sun
A tal given at University of Wisconsin at Madison April 6, 2006. RECENT PROGRESS ON CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS Zhi-Wei Sun Department of Mathematics Nanjing University Nanjing 210093,
More informationHilbert s problems, Gödel, and the limits of computation
Hilbert s problems, Gödel, and the limits of computation Logan Axon Gonzaga University November 14, 2013 Hilbert at the ICM At the 1900 International Congress of Mathematicians in Paris, David Hilbert
More informationSOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM
Acta Arith. 183(018), no. 4, 339 36. SOME VARIANTS OF LAGRANGE S FOUR SQUARES THEOREM YU-CHEN SUN AND ZHI-WEI SUN Abstract. Lagrange s four squares theorem is a classical theorem in number theory. Recently,
More informationPart II. Number Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler
More informationOn Arithmetic Properties of Bell Numbers, Delannoy Numbers and Schröder Numbers
A tal given at the Institute of Mathematics, Academia Sinica (Taiwan (Taipei; July 6, 2011 On Arithmetic Properties of Bell Numbers, Delannoy Numbers and Schröder Numbers Zhi-Wei Sun Nanjing University
More information4 Linear Recurrence Relations & the Fibonacci Sequence
4 Linear Recurrence Relations & the Fibonacci Sequence Recall the classic example of the Fibonacci sequence (F n ) n=1 the relations: F n+ = F n+1 + F n F 1 = F = 1 = (1, 1,, 3, 5, 8, 13, 1,...), defined
More informationThe Riddle of Primes
A talk given at Dalian Univ. of Technology (Nov. 16, 2012) and Nankai University (Dec. 1, 2012) The Riddle of Primes Zhi-Wei Sun Nanjing University Nanjing 210093, P. R. China zwsun@nju.edu.cn http://math.nju.edu.cn/
More informationA talk given at the University of California at Irvine on Jan. 19, 2006.
A talk given at the University of California at Irvine on Jan. 19, 2006. A SURVEY OF ZERO-SUM PROBLEMS ON ABELIAN GROUPS Zhi-Wei Sun Department of Mathematics Nanjing University Nanjing 210093 People s
More informationJournal of Combinatorics and Number Theory 1(2009), no. 1, ON SUMS OF PRIMES AND TRIANGULAR NUMBERS. Zhi-Wei Sun
Journal of Combinatorics and Number Theory 1(009), no. 1, 65 76. ON SUMS OF PRIMES AND TRIANGULAR NUMBERS Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing 10093, People s Republic of China
More informationZhi-Wei Sun Department of Mathematics, Nanjing University Nanjing , People s Republic of China
J. Number Theory 16(016), 190 11. A RESULT SIMILAR TO LAGRANGE S THEOREM Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing 10093, People s Republic of China zwsun@nju.edu.cn http://math.nju.edu.cn/
More informationCullen Numbers in Binary Recurrent Sequences
Cullen Numbers in Binary Recurrent Sequences Florian Luca 1 and Pantelimon Stănică 2 1 IMATE-UNAM, Ap. Postal 61-3 (Xangari), CP 58 089 Morelia, Michoacán, Mexico; e-mail: fluca@matmor.unam.mx 2 Auburn
More informationCounting on Chebyshev Polynomials
DRAFT VOL. 8, NO., APRIL 009 1 Counting on Chebyshev Polynomials Arthur T. Benjamin Harvey Mudd College Claremont, CA 91711 benjamin@hmc.edu Daniel Walton UCLA Los Angeles, CA 90095555 waltond@ucla.edu
More informationLecture Notes. Advanced Discrete Structures COT S
Lecture Notes Advanced Discrete Structures COT 4115.001 S15 2015-01-13 Recap Divisibility Prime Number Theorem Euclid s Lemma Fundamental Theorem of Arithmetic Euclidean Algorithm Basic Notions - Section
More informationDefining Valuation Rings
East Carolina University, Greenville, North Carolina, USA June 8, 2018 Outline 1 What? Valuations and Valuation Rings Definability Questions in Number Theory 2 Why? Some Questions and Answers Becoming
More informationZhi-Wei Sun. Department of Mathematics Nanjing University Nanjing , P. R. China
A talk given at National Cheng Kung Univ. (2007-06-28). SUMS OF SQUARES AND TRIANGULAR NUMBERS, AND RADO NUMBERS FOR LINEAR EQUATIONS Zhi-Wei Sun Department of Mathematics Nanjing University Nanjing 210093,
More informationTHE p-adic VALUATION OF LUCAS SEQUENCES
THE p-adic VALUATION OF LUCAS SEQUENCES CARLO SANNA Abstract. Let (u n) n 0 be a nondegenerate Lucas sequence with characteristic polynomial X 2 ax b, for some relatively prime integers a and b. For each
More informationDepartment of Mathematics, Nanjing University Nanjing , People s Republic of China
Proc Amer Math Soc 1382010, no 1, 37 46 SOME CONGRUENCES FOR THE SECOND-ORDER CATALAN NUMBERS Li-Lu Zhao, Hao Pan Zhi-Wei Sun Department of Mathematics, Naning University Naning 210093, People s Republic
More informationSome congruences concerning second order linear recurrences
Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae,. (1997) pp. 9 33 Some congruences concerning second order linear recurrences JAMES P. JONES PÉTER KISS Abstract. Let U n V n (n=0,1,,...) be
More informationProblems and Results in Additive Combinatorics
Problems and Results in Additive Combinatorics Zhi-Wei Sun Nanjing University Nanjing 210093, P. R. China zwsun@nju.edu.cn http://math.nju.edu.cn/ zwsun August 2, 2010 While in the past many of the basic
More informationIntroduction to Number Theory
INTRODUCTION Definition: Natural Numbers, Integers Natural numbers: N={0,1,, }. Integers: Z={0,±1,±, }. Definition: Divisor If a Z can be writeen as a=bc where b, c Z, then we say a is divisible by b or,
More informationApéry Numbers, Franel Numbers and Binary Quadratic Forms
A tal given at Tsinghua University (April 12, 2013) and Hong Kong University of Science and Technology (May 2, 2013) Apéry Numbers, Franel Numbers and Binary Quadratic Forms Zhi-Wei Sun Nanjing University
More informationOn Snevily s Conjecture and Related Topics
Jiangsu University (Nov. 24, 2017) and Shandong University (Dec. 1, 2017) and Hunan University (Dec. 10, 2017) On Snevily s Conjecture and Related Topics Zhi-Wei Sun Nanjing University Nanjing 210093,
More informationA CRITERION FOR POLYNOMIALS TO BE CONGRUENT TO THE PRODUCT OF LINEAR POLYNOMIALS (mod p) ZHI-HONG SUN
A CRITERION FOR POLYNOMIALS TO BE CONGRUENT TO THE PRODUCT OF LINEAR POLYNOMIALS (mod ) ZHI-HONG SUN Deartment of Mathematics, Huaiyin Teachers College, Huaian 223001, Jiangsu, P. R. China e-mail: hyzhsun@ublic.hy.js.cn
More informationCONGRUENCES CONCERNING LUCAS SEQUENCES ZHI-HONG SUN
Int. J. Number Theory 004, no., 79-85. CONGRUENCES CONCERNING LUCAS SEQUENCES ZHI-HONG SUN School of Mathematical Sciences Huaiyin Normal University Huaian, Jiangsu 00, P.R. China zhihongsun@yahoo.com
More informationSuper congruences involving binomial coefficients and new series for famous constants
Tal at the 5th Pacific Rim Conf. on Math. (Stanford Univ., 2010 Super congruences involving binomial coefficients and new series for famous constants Zhi-Wei Sun Nanjing University Nanjing 210093, P. R.
More informationPolygonal Numbers, Primes and Ternary Quadratic Forms
Polygonal Numbers, Primes and Ternary Quadratic Forms Zhi-Wei Sun Nanjing University Nanjing 210093, P. R. China zwsun@nju.edu.cn http://math.nju.edu.cn/ zwsun August 26, 2009 Modern number theory has
More informationBasic Algorithms in Number Theory
Basic Algorithms in Number Theory Algorithmic Complexity... 1 Basic Algorithms in Number Theory Francesco Pappalardi Discrete Logs, Modular Square Roots & Euclidean Algorithm. July 20 th 2010 Basic Algorithms
More informationarxiv: v5 [math.nt] 22 Aug 2013
Prerint, arxiv:1308900 ON SOME DETERMINANTS WITH LEGENDRE SYMBOL ENTRIES arxiv:1308900v5 [mathnt] Aug 013 Zhi-Wei Sun Deartment of Mathematics, Nanjing University Nanjing 10093, Peole s Reublic of China
More informationYi Wang Department of Applied Mathematics, Dalian University of Technology, Dalian , China (Submitted June 2002)
SELF-INVERSE SEQUENCES RELATED TO A BINOMIAL INVERSE PAIR Yi Wang Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China (Submitted June 2002) 1 INTRODUCTION Pairs of
More information198 VOLUME 46/47, NUMBER 3
LAWRENCE SOMER Abstract. Rotkiewicz has shown that there exist Fibonacci pseudoprimes having the forms p(p + 2), p(2p 1), and p(2p + 3), where all the terms in the products are odd primes. Assuming Dickson
More information(n = 0, 1, 2,... ). (2)
Bull. Austral. Math. Soc. 84(2011), no. 1, 153 158. ON A CURIOUS PROPERTY OF BELL NUMBERS Zhi-Wei Sun and Don Zagier Abstract. In this paper we derive congruences expressing Bell numbers and derangement
More informationSome new representation problems involving primes
A talk given at Hong Kong Univ. (May 3, 2013) and 2013 ECNU q-series Workshop (Shanghai, July 30) Some new representation problems involving primes Zhi-Wei Sun Nanjing University Nanjing 210093, P. R.
More informationJournal of Number Theory
Journal of Number Theory 130 (2010) 1737 1749 Contents lists available at ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt A binary linear recurrence sequence of composite numbers Artūras
More informationThe Fibonacci Quarterly 44(2006), no.2, PRIMALITY TESTS FOR NUMBERS OF THE FORM k 2 m ± 1. Zhi-Hong Sun
The Fibonacci Quarterly 44006, no., 11-130. PRIMALITY TESTS FOR NUMBERS OF THE FORM k m ± 1 Zhi-Hong Sun eartment of Mathematics, Huaiyin Teachers College, Huaian, Jiangsu 3001, P.R. China E-mail: zhsun@hytc.edu.cn
More informationPell Equation x 2 Dy 2 = 2, II
Irish Math Soc Bulletin 54 2004 73 89 73 Pell Equation x 2 Dy 2 2 II AHMET TEKCAN Abstract In this paper solutions of the Pell equation x 2 Dy 2 2 are formulated for a positive non-square integer D using
More informationOn repdigits as product of consecutive Lucas numbers
Notes on Number Theory and Discrete Mathematics Print ISSN 1310 5132, Online ISSN 2367 8275 Vol. 24, 2018, No. 3, 5 102 DOI: 10.7546/nntdm.2018.24.3.5-102 On repdigits as product of consecutive Lucas numbers
More informationCHARACTERIZING INTEGERS AMONG RATIONAL NUMBERS WITH A UNIVERSAL-EXISTENTIAL FORMULA
CHARACTERIZING INTEGERS AMONG RATIONAL NUMBERS WITH A UNIVERSAL-EXISTENTIAL FORMULA BJORN POONEN Abstract. We prove that Z in definable in Q by a formula with 2 universal quantifiers followed by 7 existential
More informationFlorian Luca Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P , Morelia, Michoacán, México
Florian Luca Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P. 8180, Morelia, Michoacán, México e-mail: fluca@matmor.unam.mx Laszlo Szalay Department of Mathematics and Statistics,
More informationA talk given at the Institute of Mathematics (Beijing, June 29, 2008)
A talk given at the Institute of Mathematics (Beijing, June 29, 2008) STUDY COVERS OF GROUPS VIA CHARACTERS AND NUMBER THEORY Zhi-Wei Sun Department of Mathematics Nanjing University Nanjing 210093, P.
More informationOn Computably Enumerable Sets over Function Fields
On Computably Enumerable Sets over Function Fields Alexandra Shlapentokh East Carolina University Joint Meetings 2017 Atlanta January 2017 Some History Outline 1 Some History A Question and the Answer
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018 57 5. p-adic Numbers 5.1. Motivating examples. We all know that 2 is irrational, so that 2 is not a square in the rational field Q, but that we can
More informationOn The Discriminator of Lucas Sequences
On The Discriminator of Lucas Sequences Bernadette Faye Ph.D Student FraZA, Bordeaux November 8, 2017 The Discriminator Let a = {a n } n 1 be a sequence of distinct integers. The Discriminator is defined
More informationON UNIVERSAL SUMS OF POLYGONAL NUMBERS
Sci. China Math. 58(2015), no. 7, 1367 1396. ON UNIVERSAL SUMS OF POLYGONAL NUMBERS Zhi-Wei SUN Department of Mathematics, Nanjing University Nanjing 210093, People s Republic of China zwsun@nju.edu.cn
More informationASSIGNMENT Use mathematical induction to show that the sum of the cubes of three consecutive non-negative integers is divisible by 9.
ASSIGNMENT 1 1. Use mathematical induction to show that the sum of the cubes of three consecutive non-negative integers is divisible by 9. 2. (i) If d a and d b, prove that d (a + b). (ii) More generally,
More informationPublic-key Cryptography: Theory and Practice
Public-key Cryptography Theory and Practice Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Chapter 2: Mathematical Concepts Divisibility Congruence Quadratic Residues
More informationResearch Statement. Enrique Treviño. M<n N+M
Research Statement Enrique Treviño My research interests lie in elementary analytic number theory. Most of my work concerns finding explicit estimates for character sums. While these estimates are interesting
More informationRelative Densities of Ramified Primes 1 in Q( pq)
International Mathematical Forum, 3, 2008, no. 8, 375-384 Relative Densities of Ramified Primes 1 in Q( pq) Michele Elia Politecnico di Torino, Italy elia@polito.it Abstract The relative densities of rational
More informationMATH 402 : 2017 page 1 of 6
ADMINISTRATION What? Math 40: Enumerative Combinatorics Who? Me: Professor Greg Smith You: students interested in combinatorics When and Where? lectures: slot 00 office hours: Tuesdays at :30 6:30 and
More informationOn the Shifted Product of Binary Recurrences
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 13 (2010), rticle 10.6.1 On the Shifted Product of Binary Recurrences Omar Khadir epartment of Mathematics University of Hassan II Mohammedia, Morocco
More informationExplicit solution of a class of quartic Thue equations
ACTA ARITHMETICA LXIV.3 (1993) Explicit solution of a class of quartic Thue equations by Nikos Tzanakis (Iraklion) 1. Introduction. In this paper we deal with the efficient solution of a certain interesting
More informationColloq. Math. 145(2016), no. 1, ON SOME UNIVERSAL SUMS OF GENERALIZED POLYGONAL NUMBERS. 1. Introduction. x(x 1) (1.1) p m (x) = (m 2) + x.
Colloq. Math. 145(016), no. 1, 149-155. ON SOME UNIVERSAL SUMS OF GENERALIZED POLYGONAL NUMBERS FAN GE AND ZHI-WEI SUN Abstract. For m = 3, 4,... those p m (x) = (m )x(x 1)/ + x with x Z are called generalized
More information= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2
8. p-adic numbers 8.1. Motivation: Solving x 2 a (mod p n ). Take an odd prime p, and ( an) integer a coprime to p. Then, as we know, x 2 a (mod p) has a solution x Z iff = 1. In this case we can suppose
More informationPROBLEMS ON CONGRUENCES AND DIVISIBILITY
PROBLEMS ON CONGRUENCES AND DIVISIBILITY 1. Do there exist 1,000,000 consecutive integers each of which contains a repeated prime factor? 2. A positive integer n is powerful if for every prime p dividing
More informationMATH 4400 SOLUTIONS TO SOME EXERCISES. 1. Chapter 1
MATH 4400 SOLUTIONS TO SOME EXERCISES 1.1.3. If a b and b c show that a c. 1. Chapter 1 Solution: a b means that b = na and b c that c = mb. Substituting b = na gives c = (mn)a, that is, a c. 1.2.1. Find
More informationTwo Diophantine Approaches to the Irreducibility of Certain Trinomials
Two Diophantine Approaches to the Irreducibility of Certain Trinomials M. Filaseta 1, F. Luca 2, P. Stănică 3, R.G. Underwood 3 1 Department of Mathematics, University of South Carolina Columbia, SC 29208;
More informationODD REPDIGITS TO SMALL BASES ARE NOT PERFECT
#A34 INTEGERS 1 (01) ODD REPDIGITS TO SMALL BASES ARE NOT PERFECT Kevin A. Broughan Department of Mathematics, University of Waikato, Hamilton 316, New Zealand kab@waikato.ac.nz Qizhi Zhou Department of
More informationp-regular functions and congruences for Bernoulli and Euler numbers
p-regular functions and congruences for Bernoulli and Euler numbers Zhi-Hong Sun( Huaiyin Normal University Huaian, Jiangsu 223001, PR China http://www.hytc.edu.cn/xsjl/szh Notation: Z the set of integers,
More informationarxiv: v1 [math.nt] 8 Jan 2014
A NOTE ON p-adic VALUATIONS OF THE SCHENKER SUMS PIOTR MISKA arxiv:1401.1717v1 [math.nt] 8 Jan 2014 Abstract. A prime number p is called a Schenker prime if there exists such n N + that p nandp a n, wherea
More informationElliptic curves and Hilbert s Tenth Problem
Elliptic curves and Hilbert s Tenth Problem Karl Rubin, UC Irvine MAA @ UC Irvine October 16, 2010 Karl Rubin Elliptic curves and Hilbert s Tenth Problem MAA, October 2010 1 / 40 Elliptic curves An elliptic
More informationHomework 8 Solutions to Selected Problems
Homework 8 Solutions to Selected Problems June 7, 01 1 Chapter 17, Problem Let f(x D[x] and suppose f(x is reducible in D[x]. That is, there exist polynomials g(x and h(x in D[x] such that g(x and h(x
More informationPredictive criteria for the representation of primes by binary quadratic forms
ACTA ARITHMETICA LXX3 (1995) Predictive criteria for the representation of primes by binary quadratic forms by Joseph B Muskat (Ramat-Gan), Blair K Spearman (Kelowna, BC) and Kenneth S Williams (Ottawa,
More informationA talk given at Suzhou Univ. (June 19) and Nankai Univ. (June 25, 2008) Zhi-Wei Sun
A talk given at Suzhou Univ. (June 19) and Nankai Univ. (June 25, 2008) AN EXTREMAL PROBLEM ON COVERS OF ABELIAN GROUPS Zhi-Wei Sun Department of Mathematics Nanjing University Nanjing 210093, P. R. China
More informationINDEFINITE QUADRATIC FORMS AND PELL EQUATIONS INVOLVING QUADRATIC IDEALS
INDEFINITE QUADRATIC FORMS AND PELL EQUATIONS INVOLVING QUADRATIC IDEALS AHMET TEKCAN Communicated by Alexandru Zaharescu Let p 1(mod 4) be a prime number, let γ P + p Q be a quadratic irrational, let
More informationAdvances in Applied Mathematics 48(2012), Constructing x 2 for primes p = ax 2 + by 2
Advances in Applied Mathematics 4(01, 106-10 Constructing x for primes p ax + by Zhi-Hong Sun School of Mathematical Sciences, Huaiyin Normal University, Huaian, Jiangsu 3001, P.R. China E-mail: zhihongsun@yahoo.com
More informationSum and shifted-product subsets of product-sets over finite rings
Sum and shifted-product subsets of product-sets over finite rings Le Anh Vinh University of Education Vietnam National University, Hanoi vinhla@vnu.edu.vn Submitted: Jan 6, 2012; Accepted: May 25, 2012;
More informationCongruent numbers via the Pell equation and its analogous counterpart
Notes on Number Theory and Discrete Mathematics ISSN 1310 5132 Vol. 21, 2015, No. 1, 70 78 Congruent numbers via the Pell equation and its analogous counterpart Farzali Izadi Department of Mathematics,
More informationRepresenting numbers as the sum of squares and powers in the ring Z n
Representing numbers as the sum of squares powers in the ring Z n Rob Burns arxiv:708.03930v2 [math.nt] 23 Sep 207 26th September 207 Abstract We examine the representation of numbers as the sum of two
More informationNotes on Continued Fractions for Math 4400
. Continued fractions. Notes on Continued Fractions for Math 4400 The continued fraction expansion converts a positive real number α into a sequence of natural numbers. Conversely, a sequence of natural
More informationArithmetic properties of lacunary sums of binomial coefficients
Arithmetic properties of lacunary sums of binomial coefficients Tamás Mathematics Department Occidental College 29th Journées Arithmétiques JA2015, July 6-10, 2015 Arithmetic properties of lacunary sums
More informationFermat s Little Theorem. Fermat s little theorem is a statement about primes that nearly characterizes them.
Fermat s Little Theorem Fermat s little theorem is a statement about primes that nearly characterizes them. Theorem: Let p be prime and a be an integer that is not a multiple of p. Then a p 1 1 (mod p).
More information2017 Number Theory Days in Hangzhou
2017 Number Theory Days in Hangzhou Saturday, Sep 23 Schedule Sheraton Grand Hangzhou Wetland Park Resort 8:30: Opening ceremony 8:45: Aleksandar Ivić: On some results and problems involving Hardy s function
More informationTheoretical Computers and Diophantine Equations *
Theoretical Computers and Diophantine Equations * Alvin Chan Raffles Junior College & K. J. Mourad Nat. Univ. of Singapore l. Introduction This paper will discuss algorithms and effective computability,
More informationQuasi-reducible Polynomials
Quasi-reducible Polynomials Jacques Willekens 06-Dec-2008 Abstract In this article, we investigate polynomials that are irreducible over Q, but are reducible modulo any prime number. 1 Introduction Let
More informationAdvanced Counting Techniques. Chapter 8
Advanced Counting Techniques Chapter 8 Chapter Summary Applications of Recurrence Relations Solving Linear Recurrence Relations Homogeneous Recurrence Relations Nonhomogeneous Recurrence Relations Divide-and-Conquer
More informationDiophantine quadruples and Fibonacci numbers
Diophantine quadruples and Fibonacci numbers Andrej Dujella Department of Mathematics, University of Zagreb, Croatia Abstract A Diophantine m-tuple is a set of m positive integers with the property that
More informationHandouts. CS701 Theory of Computation
Handouts CS701 Theory of Computation by Kashif Nadeem VU Student MS Computer Science LECTURE 01 Overview In this lecturer the topics will be discussed including The Story of Computation, Theory of Computation,
More informationIn Z: x + 3 = 2 3x = 2 x = 1 No solution In Q: 3x = 2 x 2 = 2. x = 2 No solution. In R: x 2 = 2 x = 0 x = ± 2 No solution Z Q.
THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH 1141 HIGHER MATHEMATICS 1A ALGEBRA. Section 1: - Complex Numbers. 1. The Number Systems. Let us begin by trying to solve various
More informationImpact of Women in Number Theory
Impact of Women in Number Theory Allysa Lumley Impact of Women Mathematicians on Research and Education in Mathematics BIRS 2018 March 17, 2018 Allysa Lumley (York) Impact of Women in Number Theory March
More informationDivisibility Sequences for Elliptic Curves with Complex Multiplication
Divisibility Sequences for Elliptic Curves with Complex Multiplication Master s thesis, Universiteit Utrecht supervisor: Gunther Cornelissen Universiteit Leiden Journées Arithmétiques, Edinburgh, July
More informationON VALUES OF CYCLOTOMIC POLYNOMIALS. V
Math. J. Okayama Univ. 45 (2003), 29 36 ON VALUES OF CYCLOTOMIC POLYNOMIALS. V Dedicated to emeritus professor Kazuo Kishimoto on his seventieth birthday Kaoru MOTOSE In this paper, using properties of
More informationA CONGRUENTIAL IDENTITY AND THE 2-ADIC ORDER OF LACUNARY SUMS OF BINOMIAL COEFFICIENTS
A CONGRUENTIAL IDENTITY AND THE 2-ADIC ORDER OF LACUNARY SUMS OF BINOMIAL COEFFICIENTS Gregory Tollisen Department of Mathematics, Occidental College, 1600 Campus Road, Los Angeles, USA tollisen@oxy.edu
More informationON THE POSSIBLE QUANTITIES OF FIBONACCI NUMBERS THAT OCCUR IN SOME TYPES OF INTERVALS
Acta Math. Univ. Comenianae Vol. LXXXVII, 2 (2018), pp. 291 299 291 ON THE POSSIBLE QUANTITIES OF FIBONACCI NUMBERS THAT OCCUR IN SOME TYPES OF INTERVALS B. FARHI Abstract. In this paper, we show that
More informationNOTES Edited by William Adkins. On Goldbach s Conjecture for Integer Polynomials
NOTES Edited by William Adkins On Goldbach s Conjecture for Integer Polynomials Filip Saidak 1. INTRODUCTION. We give a short proof of the fact that every monic polynomial f (x) in Z[x] can be written
More informationSOME AMAZING PROPERTIES OF THE FUNCTION f(x) = x 2 * David M. Goldschmidt University of California, Berkeley U.S.A.
SOME AMAZING PROPERTIES OF THE FUNCTION f(x) = x 2 * David M. Goldschmidt University of California, Berkeley U.S.A. 1. Introduction Today we are going to have a look at one of the simplest functions in
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More informationOn arithmetic functions of balancing and Lucas-balancing numbers
MATHEMATICAL COMMUNICATIONS 77 Math. Commun. 24(2019), 77 1 On arithmetic functions of balancing and Lucas-balancing numbers Utkal Keshari Dutta and Prasanta Kumar Ray Department of Mathematics, Sambalpur
More informationOn the number of representations of n by ax 2 + by(y 1)/2, ax 2 + by(3y 1)/2 and ax(x 1)/2 + by(3y 1)/2
ACTA ARITHMETICA 1471 011 On the number of representations of n by ax + byy 1/, ax + by3y 1/ and axx 1/ + by3y 1/ by Zhi-Hong Sun Huaian 1 Introduction For 3, 4,, the -gonal numbers are given by p n n
More informationHow many units can a commutative ring have?
How many units can a commutative ring have? Sunil K. Chebolu and Keir Locridge Abstract. László Fuchs posed the following problem in 960, which remains open: classify the abelian groups occurring as the
More informationA Note on Indefinite Ternary Quadratic Forms Representing All Odd Integers. Key Words: Quadratic Forms, indefinite ternary quadratic.
Bol. Soc. Paran. Mat. (3s.) v. 23 1-2 (2005): 85 92. c SPM ISNN-00378712 A Note on Indefinite Ternary Quadratic Forms Representing All Odd Integers Jean Bureau and Jorge Morales abstract: In this paper
More informationSPLITTING FIELDS AND PERIODS OF FIBONACCI SEQUENCES MODULO PRIMES
SPLITTING FIELDS AND PERIODS OF FIBONACCI SEQUENCES MODULO PRIMES SANJAI GUPTA, PAROUSIA ROCKSTROH, AND FRANCIS EDWARD SU 1. Introduction The Fibonacci sequence defined by F 0 = 0, F 1 = 1, F n+1 = F n
More informationBasic Algorithms in Number Theory
Basic Algorithms in Number Theory Algorithmic Complexity... 1 Basic Algorithms in Number Theory Francesco Pappalardi #2-b - Euclidean Algorithm. September 2 nd 2015 SEAMS School 2015 Number Theory and
More informationarxiv: v1 [math.co] 21 Sep 2015
Chocolate Numbers arxiv:1509.06093v1 [math.co] 21 Sep 2015 Caleb Ji, Tanya Khovanova, Robin Park, Angela Song September 22, 2015 Abstract In this paper, we consider a game played on a rectangular m n gridded
More informationPOSITIVE DEFINITE n-regular QUADRATIC FORMS
POSITIVE DEFINITE n-regular QUADRATIC FORMS BYEONG-KWEON OH Abstract. A positive definite integral quadratic form f is called n- regular if f represents every quadratic form of rank n that is represented
More informationEquivalence of Pepin s and the Lucas-Lehmer Tests
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol., No. 3, 009, (35-360) ISSN 1307-5543 www.ejpam.com Equivalence of Pepin s and the Lucas-Lehmer Tests John H. Jaroma Department of Mathematics & Physics,
More informationHilbert s Tenth Problem
Hilbert s Tenth Problem Nicole Bowen BS, Mathematics An Undergraduate Honors Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science at the University of Connecticut
More informationActa Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 19 24
Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 19 24 PRIMITIVE DIVISORS OF LUCAS SEQUENCES AND PRIME FACTORS OF x 2 + 1 AND x 4 + 1 Florian Luca (Michoacán, México) Abstract. In this
More information