On two sums related to the Lehmer problem over short intervals

<abstract><p>In this article, we study sums related to the Lehmer problem over short intervals, and give two asymptotic formulae for them. The original Lehmer problem is to count the numbers coprime to a prime such that the number and the its number theoretical inverse are in different parities in some intervals. The numbers which satisfy these conditions are called Lehmer numbers. It prompts a series of investigations, such as the investigation of the error term in the asymptotic formula. Many scholars investigate the generalized Lehmer problems and get a lot of results. We follow the trend of these investigations and generalize the Lehmer problem.</p></abstract>

A note on the ternary purely exponential diophantine equation Ax + By = Cz with A + B = C2

Let l,m,r be fixed positive integers such that 2| l, 3lm, l > r and 3 | r. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if min{rlm2 − 1,(l − r)lm2 + 1} > 30, then the equation (rlm2 − 1)x + ((l − r)lm2 + 1)y = (lm)z has only the positive integer solution (x,y,z) = (1,1,2).

Infinitude of k-Lehmer numbers which are not Carmichael

In this paper, we prove that there are infinitely many [Formula: see text]’s for which [Formula: see text] but [Formula: see text] is not a Carmichael number. Additionally, we prove that for any [Formula: see text], there exist infinitely many [Formula: see text]’s such that [Formula: see text] but [Formula: see text]. The constructions that we consider here are generalizations of Carmichael and Lehmer numbers, respectively, that were first formulated by Grau and Oller-Marcén.

A Note on the Exponential Diophantine Equation x2+q2m=2yp

ln this paper, using a deep result on the existence of primitive divisors of Lehmer numbers given by Y. Bilu, G. Hanrot and P. M. Voutier, we prove that the equation has no positive integer solution (x, y, m, p, q), where p and q are odd primes with p>3, gcd(x, y)=1 and y is not the sum of two consecutive squares.