On a compositeness test for .
On a congruence by János Bolyai, connected with pseudoprimes.
On a congruence for the number of finite marked topologies
On a congruence of Emma Lehmer related to Euler numbers
A congruence of Emma Lehmer (1938) for Euler numbers modulo p in terms of a certain sum of reciprocals of squares of integers was recently extended to prime power moduli by T. Cai et al. We generalize this further to arbitrary composite moduli n and characterize those n for which the sum in question vanishes modulo n (or modulo n/3 when 3|n). Primes for which play an important role, and we present some numerical results.
On a Conjecture of A. Ivić and W. Schwarz
On a conjecture of Bruckman.
On a conjecture of Dekking : The sum of digits of even numbers
Let and denote by the sum-of-digits function in base . For considerIn 1983, F. M. Dekking conjectured that this quantity is greater than and, respectively, less than for infinitely many , thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.
On A Conjecture of Emma Lehmer.
On a conjecture of Erdös
On a conjecture of Issai Schur.
On a conjecture of Mąkowski and Schinzel
On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions σ and ϕ
For any positive integer n let ϕ(n) and σ(n) be the Euler function of n and the sum of divisors of n, respectively. In [5], Mąkowski and Schinzel conjectured that the inequality σ(ϕ(n)) ≥ n/2 holds for all positive integers n. We show that the lower density of the set of positive integers satisfying the above inequality is at least 0.74.
On a conjecture of Narkiewicz about functions with non-decreasing normal order
On a conjecture of Norton
On a conjecture of R. L. Graham
On a conjecture of R. L. Graham
On a Conjecture of Zaremba.
On a connection of number theory with graph theory
We assign to each positive integer a digraph whose set of vertices is and for which there is a directed edge from to if . We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.
On a divisibility problem
On a divisibility problem
Let be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if , then which improves a previous result of the second author.