Assuming the validity of Dickson's conjecture, we show that the set V of values of the Euler's totient function φ contains arbitrarily large arithmetic progressions with common difference 4. This leads to the question of proving unconditionally that this set V has a positive upper Banach density.
A Piatetski-Shapiro sequence with exponent \alphais a sequence of integer parts of n^\alpha(n = 1,2,\ldots)with a non-integral \alpha > 0 We let \mathrm{PS}(\alpha)denote the set of those terms. In this article, we study the set of \alphaso that the equation ax + by = czhas infinitely many pairwise distinct solutions (x,y,z) \in \mathrm{PS}(\alpha)^3 and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many \alpha > 2such that $\mathr...
We provide explicit bounds on the difference of heights of isogenous Drinfeld modules. We derive a finiteness result in isogeny classes. In the rank 2 case, we also obtain an explicit upper bound on the size of the coefficients of modular polynomials attached to Drinfeld modules.
Let k \geq 1be a cube-free integer with k \equiv 1 \pmod {9}and \gcd(k, 7\cdot 571)=1 In this paper, we prove the existence of infinitely many triples of imaginary quadratic fields \mathbb{Q}(\sqrt{d}) \mathbb{Q}(\sqrt{d+1})and \mathbb{Q}(\sqrt{d+k^2})with d \in \mathbb{Z}such that the class number of each of them is divisible by 3 This affirmatively answers a weaker version of a conjecture of Iizuka \cite{iizuka-jnt}.
In this article we explicitly determine the structure of the Weierstrass semigroups H(P)for any point Pof the Suzuki curve \mathcal{S}_q As the point Pvaries, exactly two possibilities arise for H(P) one for the \mathbb{F}_qrational points (already known in the literature), and one for all remaining points. For this last case a minimal set of generators of H(P)is also provided. As an application, we construct dual one-point codes from an \mathbb{F}_{q^4}\setminus\fqpoint ...
We are interested in solutions of a norm form equation that takes values in a given multi-recurrence. We show that among the solutions there are only finitely many values in each component which lie in the given multi-recurrence unless the recurrence is of precisely described exceptional shape. This gives a variant of the question on arithmetic progressions in the solution set of norm form equations.
Let pbe an odd prime. We attach appropriate signed Selmer groups to an elliptic curve E where Eis assumed to have semistable reduction at all primes above p We then compare the Iwasawa \lambdainvariants of these signed Selmer groups for two congruent elliptic curves over the cyclotomic \mathbb{Z}_pextension in the spirit of Greenberg-Vatsal and B. D. Kim. As an application of our comparsion formula, we show that if the pparity conjecture is true for one of the congruent elli...