Open Conjectures on Congruences

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 conjectures of the author while conjectures in Part B have been recently confirmed. 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 offer prizes for the first correct proofs.

88 pages. For new additions, see Conj. A76-85

References in corpus (19)

• On congruences related to central binomial coefficients
• New congruences involving products of two binomial coefficients
• On Delannoy numbers and Schröder numbers
• Various congruences involving binomial coefficients and higher-order Catalan numbers
• On sums involving products of three binomial coefficients
• On sums of binomial coefficients modulo p^2
• Congruences involving the reciprocals of central binomial coefficients
• p-adic valuations of some sums of multinomial coefficients
• Fibonacci numbers modulo cubes of primes
• Supercongruences for a truncated hypergeometric series
• Some congruences for the second-order Catalan numbers
• A new series for ${\pi }^3$ and related congruences
• On Apery numbers and generalized central trinomial coefficients
• Arithmetic theory of harmonic numbers (II)
• Congruences involving binomial coefficients and Lucas sequences
• Proof of three conjectures on congruences
• Curious congruences for Fibonacci numbers
• On divisibility concerning binomial coefficients
• Some congruences involving binomial coefficients

Cited by in corpus (36)

• Dwork-type supercongruences through a creative $q$-microscope
• Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
• On sums of Apéry polynomials and related congruences
• Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+d{y}^2$
• Some new series for $1/\pi$ and related congruences
• Supercongruences on some binomial sums involving Lucas sequences
• Proof of a congruence for harmonic numbers conjectured by Z.-W. Sun
• Products and sums divisible by central binomial coefficients
• A rapidly converging Ramanujan-type series for Catalan's constant
• Proof of two supercongruences by the Wilf-Zeilberger method
• Some supercongruences modulo $p^2$
• On two conjectural supercongruences of Z.-W. Sun
• Generalized Legendre polynomials and related congruences modulo $p^2$
• Congruences involving ${\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{4k}{2k}\right)m^{-k}$
• On divisibility concerning binomial coefficients
• Supercongruences for central trinomial coefficients
• Two congruences concerning Apéry numbers
• Congruences for Domb and Almkvist-Zudilin numbers
• Determining $x$ or $y$ mod $p^2$ with $p=x^2+d{y}^2$
• New congruences for sums involving Apery numbers or central Delannoy numbers
• Three supercongruences for Apery numbers or Franel numbers
• $p$-adic analogues of hypergeometric identities and their applications
• Note on super congruences modulo $p^2$
• A parametric congruence motivated by Orr's identity
• Divisibility of some binomial sums
• On two congruences involving Franel numbers
• Supercongruences for sums involving Domb numbers
• $p$-adic supercongruences conjectured by Sun
• Proof of a supercongruence conjectured by Sun through a $q$-microscope
• Some new series for $1/\pi$ motivated by congruences
• Proof of some conjectural hypergeometric supercongruences via curious identities
• New type series for powers of $\pi$
• On a conjecture related to integer-valued polynomials
• Proof of a conjectural supercongruence modulo $p^5$
• Proof of Sun's supercongruence involving Catalan numbers
• Some Numeric Hypergeometric Supercongruences