On a Conjecture of Hecke Concerning Elementary Class Number Formulas.
On a conjecture of Issai Schur.
On a conjecture of Izhboldin on similarity of quadratic forms.
On a conjecture of Lemke and Kleitman
On a Conjecture of Littlewood Concerning Exponential Sums, I
On a Conjecture of Littlewood Concerning Exponentials Sums, II
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 Morgan Ward, I
On a conjecture of Morgan Ward, II
On a conjecture of Morgan Ward, III
On a conjecture of Narkiewicz about functions with non-decreasing normal order
On a conjecture of Norton
On a conjecture of Pomerance
On a conjecture of R. L. Graham
On a conjecture of R. L. Graham
On a conjecture of Sárközy and Szemerédi
Two infinite sequences A and B of non-negative integers are called infinite additive complements if their sum contains all sufficiently large integers. In 1994, Sárközy and Szemerédi conjectured that there exist infinite additive complements A and B with lim sup A(x)B(x)/x ≤ 1 and A(x)B(x)-x = O(minA(x),B(x)), where A(x) and B(x) are the counting functions of A and B, respectively. We prove that, for infinite additive complements A and B, if lim sup A(x)B(x)/x ≤ 1, then, for any given M > 1,...
On a Conjecture of Siegel.
On a conjecture of Watkins
Watkins has conjectured that if is the rank of the group of rational points of an elliptic curve over the rationals, then divides the modular parametrisation degree. We show, for a certain class of , chosen to make things as easy as possible, that this divisibility would follow from the statement that a certain -adic deformation ring is isomorphic to a certain Hecke ring, and is a complete intersection. However, we show also that the method developed by Taylor, Wiles and others, to prove...
On a conjecture of Yiming Long