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Let p be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for P p−1 2 k=0 2k k 2 m−k (mod p2). In particular, we confirm se...
Proofs of power sum and binomial coefficient congruences via Pascal's identity
binomial coefficient Pascal's identity
2010/11/8
A frequently cited theorem says that for n > 0 and prime p, the sum of the first p n-th powers is congruent to -1 modulo p if p-1 divides n, and to 0 otherwise. We survey the main ingredients in seve...
Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences
Andrews' spt-function Atkin's Hecke-type partition congruences
2010/11/15
New congruences are found for Andrews' smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part alpha(24z) of a certain weak Maass form M(z) of ...
Congruences for Andrews' spt-function modulo powers of 5, 7 and 13
Andrews' spt-function math
2010/11/15
Congruences are found modulo powers of 5, 7 and 13 for Andrews' smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan's partition congruences modulo powers of 5, 7 a...
Some new super-congruences modulo prime powers
super-congruences modulo prime powers math
2010/11/22
Let $p>3$ be a prime. We show that $$\sum_{k=0}^{p-1}\binom(-1/(p+1),k)^{p+1}=0 (mod p^5).$$ For any positive integer m not divisible by p, we prove that $$\sum_{k=0}^{p-1}(-1)^{km}\binom(p/m-1,k)^m=0...
Inhomogeneous cubic congruences and rational points on del Pezzo surfaces
Inhomogeneous cubic congruences rational points del Pezzo surfaces
2010/11/22
For given non-zero integers a,b,q we investigate the density of integer solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q) and use it to establish the Manin conjecture for a singular d...
Power multiples in binary recurrence sequences: an approach by congruences
Power multiples binary recurrence sequences approach by congruences
2010/12/13
We introduce an elementary congruence-based procedure to look for q-th power multiples in arbitrary binary recurrence sequences (q 3). The procedure allows to prove that no such multiples exist in m...
A new series for $\pi^3$ and related congruences
Series for 3 central binomial coefficients congruences modulo prime powers
2010/12/14
Let H(2)n denote the second-order harmonic numberP0or n = 0, 1, 2, . . . . In this paper we obtain the following new identity for 3: 1 X k=1 2kH(2) k−1 k2k k = 3 48 .We e...
Congruences on the Bell polynomials and the derangement polynomials
Bell polynomials Derangement polynomials Stirling numbers Congruences
2010/12/7
In this note, by the umbra calculus method, the Sun and Zagier’s congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences ar...
A notion of congruence on fuzzy algebras is introduced and the isomorphism theorem is established.
A famous result of Hausdorff states that a sphere with countably many points removed can be partitioned into three pieces A,B,C such that A is congruent to B (i.e., there is an isometry of the sphere...
Solutions to congruences using sets with the property of Baire
Solutions sets property of Baire
2010/10/29
Hausdorff's paradoxical decomposition of a sphere with countably many points removed (the main precursor of the Banach-Tarski paradox) actually produced a partition of this set into three pieces A,B,...
Some Results on Space-Like Line Congruences and Their Space-Like Parameter Ruled Surface
Space-Like Line Congruences Space-Like Parameter Ruled Surface
2010/3/3
Using dual vectors, space-like ruled surfaces was introduced in [2]. In this paper, we define space-like line congruence and their space-like parameter ruled surfaces {\mathbf R11(t)} and {\mathbf R21...