PDF《arXiv:0911.5665》

arXiv:0911.

5665v57 [math.NT]

24 Aug

2011 arXiv:0911.5665 OPEN CONJECTURES ON CONGRUENCES Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing 210093, People'

s Republic of China zwsun@nju.edu.cn http://math.nju.edu.cn/?zwsun Abstract. We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of

100 unsolved con- jectures of the author while conjectures in Part B have been recently con- ?rmed. We hope that this material will interest number theorists and stimulate further research. Number theorists are welcome to work on those open conjectures;

for some of them we o?er prizes for the ?rst correct proofs. Introduction Congruences modulo primes have been widely investigated since the time of Fermat. However, we ?nd that there are still lots of new challenging congruences that cannot be easily solved. They appeal for new powerful tools or advanced theory. Here we collect various conjectures of the author on congruences, most of which can be found in the author'

s papers available from arxiv or his homepage. We use two sections to state conjectures and related re- marks. Part A contains

100 unsolved conjectures of the author while Part B consists of conjectures that have been recently con?rmed. Most of the congruences here are super congruences in the sense that they happen to hold modulo some higher power of p. The topic of super congruences is related to the p-adic Γ-function, Gauss and Jacobi sums, hypergeomet- ric series, modular forms, Calabi-Yau manifolds, and some sophisticated combinatorial identities involving harmonic numbers (cf. K. Ono'

s book [O]). The recent theory of super congruences also involves Bernoulli and

2010 Mathematics Subject Classi?cation. Primary 11B65, 11A07;

Secondary 05A10, 05A15, 11A41, 11B39, 11B68, 11E25, 11S99, 33C20, 65B10. Keywords. Binomial coe?cients, Catalan numbers, Bernoulli numbers, Euler num- bers, series involving π, binary quadratic forms, congruences modulo prime powers. c Copyright is owned by the author Zhi-Wei Sun. The material on the author'

s homepage has been linked to Number Theory Web since Nov. 27, 2009.

1 2 ZHI-WEI SUN Euler numbers (see [S11b, S11e]) and various series related to π (cf. [vH], [S-7] and [T1]). Many congruences collected here are about p?1 k=0 ak/mk modulo powers of a prime p, where m is an integer not divisible by p and the quantity ak is a sum or a product of some binomial coe?cients which usually arises from enumerative combinatorics. For clarity, we often state the prime version of a conjecture instead of the general version. We also include several challenging conjectures on series related to π or the Dirichlet L-function which arose from the author'

s investigations of congruences, see Conjectures A3, A5, A40, A44. For some conjectures we announce prizes for the ?rst correct proofs, see remarks after Conjectures A1, A39, A44, A46, A48. Now we introduce some basic notation in this paper. As usual, we set N = {0, 1, 2,and Z+ = {1, 2, 3, The Kronecker symbol δm,n takes

1 or

0 according as m = n or not. The rising factorial (x)n is de?ned by (x)n = n?1 k=0 (x+k), and (x)0 is regarded as 1. For an integer m and a positive odd number n, the notation (m n ) stands for the Jacobi symbol. For an odd prime p, we use qp(2) to denote the Fermat quotient (2p?1 ? 1)/p. For a prime p and a rational number x, the p-adic valuation of x is given by νp(x) = sup{a ∈ N : x ≡