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We exploit the properties of Legendre polynomials defined by the contour integral $𝐏_{n} (z) = {(2 π i)}^{- 1} \oint {(1 - 2 t z + t^{2})}^{- 1 / 2} t^{- n - 1} d t,$ where the contour encloses the origin and is traversed in the counterclockwise direction, to obtain congruences of certain sums of central binomial coefficients. More explicitly, by comparing various expressions of the values of Legendre polynomials, it can be proved that for any positive integer $r$ , a prime $p ⩾ 5$ and $n = r p^{2} - 1$ , we have $\sum_{k = 0}^{⌊ n / 2 ⌋} (\binom{2 k}{k}) \equiv 0, 1 or - 1 (mod p^{2})$ , depending on the value of $r (mod 6)$ .

Congruences for certain families of Apéry-like sequences

Zhi-Hong Sun (2022)

Czechoslovak Mathematical Journal

We systematically investigate the expressions and congruences for both a one-parameter family ${G_{n} (x)}$ as well as a two-parameter family ${G_{n} (r, m)}$ of sequences.

Congruences for m-ary partitions.

Gunnar Dirdal (1975)

Mathematica Scandinavica

Congruences for $q^{[p / 8]} (m o d p)$

Zhi-Hong Sun (2013)

Acta Arithmetica

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine $q^{[p / 8]} (m o d p)$ in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.

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Composite covering systems of minimum cardinality.

Computational results relating to problems concerning $σ (n)$

Computo del numero delle coppie di numeri primi gemelli comprese tra $100.000$ e $1.100.000$ , distinte secondo le cifre terminali.

Concerning a conjecture of R. L. Graham

Concours d'admission à l'École normale supérieure en 1902

Congruence and uniqueness of certain Markoff numbers

Congruence properties for a class of arithmetical functions.

Congruence properties of coefficients of certain algebraic power series

Congruence properties of Tchebycheff polynomials

Congruence properties of the number of solutions of some equations.

Congruences for ${(A + \sqrt (A ² + m B ²))}^{(p - 1) / 2)}$ and ${(b + \sqrt (a ² + b ²))}^{(p - 1) / 4} (m o d p)$

Congruences for a class of alternating lacunary sums of binomial coefficients.

Congruences for certain binomial sums

Congruences for certain families of Apéry-like sequences

Congruences for m-ary partitions.

Congruences for $q^{[p / 8]} (m o d p)$

Congruences for representations of primes by binary quadratic forms

Congruences for Stirling numbers and Eulerian numbers

Congruences for the Convolution of Divisor sum function

Congruences for the partition function modulo powers of 5

Currently displaying 41 – 60 of 110