A p-ADIC SUPERCONGRUENCE CONJECTURE OF VAN HAMME

By Eric T. Mortenson. 2008

127 citations recorded by the source corpus.

. In this paper we prove a p-adic supercongruence conjecture of van Hamme by placing it in the context of the Beukers-like supercongruences of Rodriguez-Villegas. This conjecture is a p-adic analog of a formula of Ramanujan.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/d1fa9206a5ea38db665953dd33b6311e945a04b1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

On a q-deformation of modular forms (2018, unknown): “Mortenson [20] (see also [29]).”
Hypergeometric evaluation identities and supercongruences (2009, methodology): “A new proof of (1) We briefly outline our method for proving the next few supercongruences; we are motivated by [McCarthy and Osburn 2008] and [Mortenson 2008].”
Super congruences and Euler numbers (2011, background): “k ≡ −1 p p (mod p4) for p > 3, (p−X1)/2 k=0 (6k + 1) 2k k 3 (−512)k ≡ −2 p p (mod p3), (p−X1)/2 k=0 (42k + 5) 2k k 3 4096k ≡ −1 p 5p (mod p4). The ﬁrst of these was proved by Mortenson [M08] and the second one was recently shown by L. Long [Lo]; the third and the fourth remain open. Our fol”
Some congruences involving fourth powers of central q-binomial coefficients (2019, result): “In 1997, van Hamme [23] conjectured that, for any prime p 3, ((p−1)/(2))∑ k=0 4k + 1 (−64)k ( 2k k )3 ≡ p(−1)((p−1)/(2)) (mod p3), (1.1) which was later proved by Mortenson [14] and Zudilin [26].”
A q -analogue of a Ramanujan-type supercongruence involving central binomial coefficients (2018, methodology): “[23] E. Mortenson, A p-adic supercongruence conjecture of van Hamme, Proc.”
Super congruences and Euler numbers (2010, background): “k ≡ −1 p p (mod p4) for p > 3, (p−X1)/2 k=0 (6k + 1) 2k k 3 (−512)k ≡ −2 p p (mod p3), (p−X1)/2 k=0 (42k + 5) 2k k 3 4096k ≡ −1 p 5p (mod p4). The ﬁrst of these was proved by Mortenson [M08] and the second one was recently shown by L. Long [Lo]; the third and the fourth remain open. Our fol”
On the (K.2) supercongruence of Van Hamme (2015, background): “Mortenson [10] then used a similar argument to deal with (B.”
Supercongruences on Truncated Hypergeometric Series (2015, unknown): “2)] and first proved by Mortenson [8].”
Congruences concerning Legendre polynomials (2010, unknown): “Corollary 3.2. Let p be an odd prime. Then p−1 X2 k=0 k 2k k 3 (−64)k ≡ (−1)p−1 2 p 4 (mod p2) for p ≡ 5,7 (mod 8) and p−1 X2 k=0 k 2k k 3 256k ≡ (−1)p−1 2 p 6 (mod p2) for p ≡ 2 (mod 3). Proof. From [M2] we know that p−1 X2 k=0 (4k +1) 2k k 3 (−64)k ≡ (−1)p−1 2 p (mod p3). From [L] we have p−1 X2 k”
A q-microscope for supercongruences (2018, methodology): “Development of methods [7, 20–22, 24, 25, 27, 31, 32, 34] for establishing Ramanujan-type supercongruences like (2) and (3), sometimes modulo a smaller power of p and normally on a case-by-case study, was mainly hypergeometric.”
Proof of a q-supercongruence conjectured by Guo and Schlosser (2020, methodology): “n’s and Ramanujan-like formulas for 1/π. In particular, the following supercongruence of Van Hamme [17, (B.2)], (pX−1)/2 k=0 4k +1 (−64)k 2k k 3 ≡ p(−1)p−1 2 (mod p3) was ﬁrst proved by Mortenson [14] using a 6F 5 transformation and a technical evaluation of a quotient of Gamma functions, where p ”
Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+dy^2$ (2011, background): “dentity (of Ramanujan’s type) X∞ k=0 4k +1 (−64)k 2k k 3 = 2 π he conjectured that pX−1 k=0 4k + 1 (−64)k 2k k 3 ≡ p −1 p (mod p3) for any odd prime p (and this was conﬁrmed by E. Mortenson [M]). The reader may consult [BBC], [Be], [BoBo] and [ChCh] for more Ramanujan-type series for 1/π. The au”