{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:E5I3QP7FKEPH2PWSYBIOMSU77X","short_pith_number":"pith:E5I3QP7F","schema_version":"1.0","canonical_sha256":"2751b83fe5511e7d3ed2c050e64a9ffdec0d34776a633150a0ac5c21cf30c488","source":{"kind":"arxiv","id":"1712.02095","version":2},"attestation_state":"computed","paper":{"title":"On binomial coefficients modulo squares of primes","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Darij Grinberg","submitted_at":"2017-12-06T09:14:40Z","abstract_excerpt":"We give elementary proofs for the Apagodu-Zeilberger-Stanton-Amdeberhan-Tauraso congruences $$\\sum\\limits_{n=0}^{p-1}\\dbinom{2n}{n} \\equiv\\eta_{p}\\mod p^{2},$$ $$\\sum\\limits_{n=0}^{rp-1}\\dbinom{2n}{n} \\equiv\\eta_{p}\\sum\\limits_{n=0}^{r-1}\\dbinom {2n}{n}\\mod p^{2}$$ and $$\\sum\\limits_{n=0}^{rp-1}\\sum\\limits_{m=0}^{sp-1}\\dbinom{n+m}{m}^{2} \\equiv\\eta_{p} \\sum\\limits_{m=0}^{r-1}\\sum\\limits_{n=0}^{s-1}\\dbinom{n+m}{m}^2\\mod p^2,$$ where $p$ is an odd prime, $r$ and $s$ are nonnegative integers, and $\\eta_{p}= \\begin{cases} 0, &\\text{if }p\\equiv0\\mod 3;\\\\ 1, & \\text{if }p\\equiv1\\mod 3;\\\\ -1, 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binomial coefficients modulo squares of primes","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Darij Grinberg","submitted_at":"2017-12-06T09:14:40Z","abstract_excerpt":"We give elementary proofs for the Apagodu-Zeilberger-Stanton-Amdeberhan-Tauraso congruences $$\\sum\\limits_{n=0}^{p-1}\\dbinom{2n}{n} \\equiv\\eta_{p}\\mod p^{2},$$ $$\\sum\\limits_{n=0}^{rp-1}\\dbinom{2n}{n} \\equiv\\eta_{p}\\sum\\limits_{n=0}^{r-1}\\dbinom {2n}{n}\\mod p^{2}$$ and $$\\sum\\limits_{n=0}^{rp-1}\\sum\\limits_{m=0}^{sp-1}\\dbinom{n+m}{m}^{2} \\equiv\\eta_{p} \\sum\\limits_{m=0}^{r-1}\\sum\\limits_{n=0}^{s-1}\\dbinom{n+m}{m}^2\\mod p^2,$$ where $p$ is an odd prime, $r$ and $s$ are nonnegative integers, and $\\eta_{p}= \\begin{cases} 0, &\\text{if }p\\equiv0\\mod 3;\\\\ 1, & \\text{if }p\\equiv1\\mod 3;\\\\ -1, 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