Open conjectures on congruences. (English) Zbl 1449.11001
Summary: We collect here 100 open conjectures on congruences made by the author, some of which have never been published. This is a new edition of the author’s preprint [arXiv:0911.5665] with those confirmed conjectures removed and some new conjectures added. Many congruences here are related to representations of primes by binary quadratic forms or series for powers of \(\pi\); for example, we mention two new conjectural identities
\[\sum_{n=0}^\infty {\frac{12n+1}{100^n}} \binom{2n}{n} \sum_{k=0}^n \binom{2k}{k}^2 \binom{2(n-k)}{n-k}\left (\frac{9}{4}\right)^{n-k} = \frac{75}{4\pi}\]
and
\[\sum_{k=1}^\infty \frac{3H_{k-1}^2 + 4H_{k-1}/k}{k^2\binom{2k}{k}} = \frac{\pi^4}{360} { with }H_{K-1}: = \sum_{0 < j \le k-1} \frac{1}{j}\]
and include related congruences. We hope that this paper will interest number theorists and stimulate further research.
\[\sum_{n=0}^\infty {\frac{12n+1}{100^n}} \binom{2n}{n} \sum_{k=0}^n \binom{2k}{k}^2 \binom{2(n-k)}{n-k}\left (\frac{9}{4}\right)^{n-k} = \frac{75}{4\pi}\]
and
\[\sum_{k=1}^\infty \frac{3H_{k-1}^2 + 4H_{k-1}/k}{k^2\binom{2k}{k}} = \frac{\pi^4}{360} { with }H_{K-1}: = \sum_{0 < j \le k-1} \frac{1}{j}\]
and include related congruences. We hope that this paper will interest number theorists and stimulate further research.
MSC:
11-02 | Research exposition (monographs, survey articles) pertaining to number theory |
11A07 | Congruences; primitive roots; residue systems |
11A41 | Primes |
05A10 | Factorials, binomial coefficients, combinatorial functions |
11B65 | Binomial coefficients; factorials; \(q\)-identities |
11B68 | Bernoulli and Euler numbers and polynomials |
Keywords:
open conjectures; congruences; binomial coefficients; Bernoulli numbers; Euler numbers; series involving \(\pi\); binary quadratic formsOnline Encyclopedia of Integer Sequences:
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