Partitioning multipartite numbers into a fixed number of parts which satisfy congruence conditions.
Pascal's triangle (mod 9)
Periodic sequences of pseudoprimes connected with Carmichael numbers and the least period of the function
Petites solutions des congruences
Poly-Bernoulli numbers
By using polylogarithm series, we define “poly-Bernoulli numbers” which generalize classical Bernoulli numbers. We derive an explicit formula and a duality theorem for these numbers, together with a von Staudt-type theorem for di-Bernoulli numbers and another proof of a theorem of Vandiver.
Polynomial extension of Fleck's congruence
Positive integers such that .
Poznámka o dělitelnosti čísel
Primes, permutations and primitive roots.
Primitive Points on a Modular Hyperbola
For positive integers m, U and V, we obtain an asymptotic formula for the number of integer points (u,v) ∈ [1,U] × [1,V] which belong to the modular hyperbola uv ≡ 1 (mod m) and also have gcd(u,v) =1, which are also known as primitive points. Such points have a nice geometric interpretation as points on the modular hyperbola which are "visible" from the origin.
Primitive roots and powers among values of polynomials over finite fields.
Primitive roots and quadratic non-residues
Příspěvek k nauce o číslech
Příspěvek k nauce o číslech
Příspěvek k reducibilitě binomických kongruencí
Products of binomial coefficients modulo p²
Products of factorials modulo p
We show that if p ≠ 5 is a prime, then the numbers cover all the nonzero residue classes modulo p.
Proof of the quadratic reciprocity law in primitive recursive arithmetic.
Pseudoprime Cullen and Woodall numbers
We show that if a > 1 is any fixed integer, then for a sufficiently large x>1, the nth Cullen number Cₙ = n2ⁿ +1 is a base a pseudoprime only for at most O(x log log x/log x) positive integers n ≤ x. This complements a result of E. Heppner which asserts that Cₙ is prime for at most O(x/log x) of positive integers n ≤ x. We also prove a similar result concerning the pseudoprimality to base a of the Woodall numbers given by Wₙ = n2ⁿ - 1 for all n ≥ 1.
-analogues of two supercongruences of Z.-W. Sun
We give several different -analogues of the following two congruences of Z.-W. Sun: where is an odd prime, is a positive integer, and is the Jacobi symbol. The proofs of them require the use of some curious -series identities, two of which are related to Franklin’s involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012.