Ramanujan-type supercongruences

By Wadim Zudilin. 2008

199 citations recorded by the source corpus.

Abstract We present several supercongruences that may be viewed as p-adic analogues of Ramanujan-type series for 1 / π and 1 / π 2 , and prove three of these examples.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/d365fb45795c6398f0d21a06d91abd7193bdb207

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Super congruences and Euler numbers (2010, background): “S 31 Motivated by the Ramanujan series X∞ k=0 (20k + 3) 2k k 2 4k 2k (−210)k = 8 π and its p-adic analogue pX−1 k=0 (20k +3) 2k k 2 4k 2k (−210)k ≡ 3p −1 p (mod p3) (for odd primes p) obtained in [Zu] which absorbed the author’s attention quite recently, we ﬁnd the following conjecture. Conjecture”
Some congruences involving fourth powers of central q-binomial coefficients (2019, methodology): “In a previous paper, motivated by Zudilin’s work [26], the first author [8] used the q-WZ method to prove that, for any odd prime p, ((p−1)/(2))∑ k=0 (−1)kqk2 [4k + 1] (q; q 2)3k (q2; q2)3k ≡ [p]q(((p−1)2)/(4))(−1)((p−1)/(2)) (mod [p]3), (1.2) where the q-shifted factorial is defined by (a; q)0 = 1 ”
Super congruences and Euler numbers (2011, background): “S 31 Motivated by the Ramanujan series X∞ k=0 (20k + 3) 2k k 2 4k 2k (−210)k = 8 π and its p-adic analogue pX−1 k=0 (20k +3) 2k k 2 4k 2k (−210)k ≡ 3p −1 p (mod p3) (for odd primes p) obtained in [Zu] which absorbed the author’s attention quite recently, we ﬁnd the following conjecture. Conjecture”
A q -analogue of a Ramanujan-type supercongruence involving central binomial coefficients (2018, methodology): “Zudilin [34] has proved more Ramanujan-like supercongruences by using the WZpairs borrowed from Guillera [9, 10].”
Some congruences related to a congruence of Van Hamme (2019, background): “The supercongruence (1.1) was first proved by Mortenson [3] in 2008 and reproved by Zudilin [4] in 2009.”
Open Conjectures on Congruences (2009, methodology): “hich case 2an is an odd integer. Remark. It is known (cf. [G5]) that X∞ k=0 (14151k +827)(−1)k(1/2)k(1/3)k(2/3)k k!35002k = X∞ k=0 14151k +827 (−300)3k 2k k 2 4k 2k = 1500 √ 3 π and W. Zudilin [Zu] suggested that for any prime p > 5 we have pX−1 k=0 14151k +827 (−300)3k 2k k 2 3k k ≡ 827p p”
Hypergeometric Series, Truncated Hypergeometric Series, and Gaussian Hypergeometric Functions (2015, background): “riety of diﬀerent techniques for proving such results and each has its own strength. See [9] for another Women in Numbers (WIN) project on supercongruences, which was motivated by the work of Zudilin [39] and his conjectures. We prove Theorems 6 and 7 by deforming truncated hypergeometric series usi”
On the supercongruence conjectures of van Hamme (2015, methodology): “The supercongruence (B.2) has been proved in three ways, by Mortenson [13] using a technical evaluation of a quotient of Gamma functions, by Zudilin [18] using the W-Z method, and by Long [9] using hypergeometric series identities and evaluations.”
A q-microscope for supercongruences (2018, methodology): “Part of the arithmetic story, which is the main topic of the present note, is a production of Ramanujan-type supercongruences [34] (with some particular instances indicated in the earlier work [32] of Van Hamme): truncation of a Ramanujan-type infinite sum at the (p−1)-th place happens to be a simpl”
Some supercongruences occurring in truncated hypergeometric series (2014, unknown): “We thank Heng Huat Chan for conversations which led to §2.2, Wadim Zudilin for inspiring discussions and the reference [27] and Jonas Kibelbek for his comments on an earlier version.”
Divergent Ramanujan-type supercongruences (2010, methodology): “(12) We find quite illogical that our strategy based on the creative Wilf–Zeilberger theory [11] of WZ-pairs allows us to do only three entries from the list (1)–(9); a very similar lack of luck was reported in [19].”
On the (K.2) supercongruence of Van Hamme (2015, methodology): “4) The functions (see [4] or [16]) F (n, k) = (84n(2) − 56nk + 4k(2) + 52n− 12k + 5) (−1)k((1) 2)n( 1 2)n+k( 1 2 ) 2 n−k 2(1)(2)n(1)2n−k+1 and G(n, k) = 64n(2) (−1)k((1) 2 )n( 1 2 )n+k−1( 1 2) 2 n−k 2(1)(2)n(1)2n−k”
On Van Hamme's (A.2) and (H.2) supercongruences (2018, methodology): “It is known that Gaussian hypergeometric series (see, for example, [3–6, 12]) and the W-Z method (see, for example, [2, 8, 11, 15]) commonly apply to the Ramanujan-type supercongruences.”
A family of q-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial (2019, background): “For some background on Ramanujan-type supercongruences, the reader is referred to Zudilin’s paper [35].”