On divisibility of Narayana numbers by primes.
On divisibility of one special type of numbers
On exactly covering systems of congruences having moduli occurring at most twice
On exceptional systems of random integers.
On Giuga's Conjecture.
On Grosswald's conjecture on primitive roots
Grosswald’s conjecture is that g(p), the least primitive root modulo p, satisfies g(p) ≤ √p - 2 for all p > 409. We make progress towards this conjecture by proving that g(p) ≤ √p -2 for all and for all .
On iteration digraph and zero-divisor graph of the ring
In the first part, we assign to each positive integer a digraph whose set of vertices consists of elements of the ring with the addition and the multiplication operations modulo and for which there is a directed edge from to if and only if . Associated with are two disjoint subdigraphs: and whose union is The vertices of are coprime to and the vertices of are not coprime to In this part, we study the structure of in detail. In the second part, we investigate the zero-divisor...
On Krasner's Criteria for the First Case of Fermat's Last Theorem.
On k-triad sequences.
On Lucas pseudoprimes of the form in arithmetic progression with a prescribed value of the Jacobi symbol
On m times covering systems of congruences
On mod logarithms and .
On Newman's conjecture and prime trees.
On Pascal’s triangle modulo
On power residues
On primitive roots.
On properties of systems of arithmetic sequences
On pseudoprimes having special forms and a solution of K. Szymiczek’s problem
We use the properties of -adic integrals and measures to obtain general congruences for Genocchi numbers and polynomials and tangent coefficients. These congruences are analogues of the usual Kummer congruences for Bernoulli numbers, generalize known congruences for Genocchi numbers, and provide new congruences systems for Genocchi polynomials and tangent coefficients.
On q-Euler numbers, q-Salié numbers and q-Carlitz numbers
On restricted m-ary partitions.