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.

We study so-called real zeros of holomorphic Hecke cusp forms, that is, zeros on three geodesic segments on which the cusp form (or a multiple of it) takes real values. Ghosh and Sarnak, who were the first to study this problem, showed the existence of many such zeros if many short intervals contain numbers whose prime factors all belong to a certain subset of the primes.We prove new results concerning this sieving problem which leads to improved lower bounds for the number of real zeros.

Soient $\lambda \>1,0\<\eta \<\frac{1}{2}$ et $g\left(n\right)$ une fonction multiplicative vérifiant $\left\{\begin{array}{c}g\left(p\right)=1/\lambda \\ g\left(n\right)\gg n^{-\eta }\end{array}\right.$. Dans ce travail, on établit une formule asymptotique de la somme ${\sum }_{ng\left(n\right)\le x;P\left(n\right)\le y}1$, valable dans le domaine exp${(loglogcx)}^{\frac{5}{3}+ϵ}\le y/\lambda \le cx$, et on donne une condition nécessaire et suffisante pour que cette somme soit équivalente à ${\sum }_{n\le x;P\left(n\right)\le y}1/g\left(n\right)$.

This paper studies integer solutions to the $abc$ equation $A+B+C=0$ in which none of $A,B,C$ have a large prime factor. We set $H(A,B,C)=max\left(\right|A|,|B|,|C\left|\right)$, and consider primitive solutions ($\gcd (A,B,C)=1$) having no prime factor larger than ${(logH(A,B,C))}^{\kappa }$, for a given finite $\kappa$. We show that the $abc$ Conjecture implies that for any fixed $\kappa \<1$ the equation has only finitely many primitive solutions. We also discuss a conditional result, showing that the Generalized Riemann hypothesis (GRH) implies that for any fixed $\kappa \>8$ the $abc$ equation has infinitely many primitive solutions....

We employ a generalised version of Heath-Brown's square sieve in order to establish an asymptotic estimate of the number of solutions a, b ∈ ℕ to the equations a + b = n and a - b = n, where a is k-free and b is l-free. This is the first time that this problem has been studied with distinct powers k and l.