arXiv subjects

Enrique González-Jiménez

Publications and source records attributed to Enrique González-Jiménez.

21 records · Page 2Linked to original sources

arXiv2013

Five squares in arithmetic progression over quadratic fields

We give several criteria to show over which quadratic number fields Q(sqrt{D}) there should exists a non-constant arithmetic progressions of five squares. This is done by translating the problem to determining when some genus five curves C_D defined over Q have rational points, and then using a Mordell-Weil sieve argument among others. Using a elliptic Chabauty-like method, we prove that the only non-constant arithmetic progressions of five squares over Q(sqrt{409}), up to equivalence, is 7^2, 13^2, 17^2, 409, 23^2. Furthermore, we give an algorithm that allow to construct all the non-constant arithmetic progressions of five squares over all quadratic fields. Finally, we state several problems and conjectures related to this problem.

math.NT↗

arXiv2013

On a conjecture of Rudin on squares in Arithmetic Progressions

Let Q(N;q,a) denotes the number of squares in the arithmetic progression qn+a, for n=0, 1,...,N-1, and let Q(N) be the maximum of Q(N;q,a) over all non-trivial arithmetic progressions qn + a. Rudin's conjecture asserts that Q(N)=O(Sqrt(N)), and in its stronger form that Q(N)=Q(N;24,1) if N=> 6. We prove the conjecture above for 6<=N<=52. We even prove that the arithmetic progression 24n+1 is the only one, up to equivalence, that contains Q(N) squares for the values of N such that Q(N) increases, for 7<=N<=52 (hence, for N=8,13,16,23,27,36,41 and 52). This allow us to assert, what we have called Super-Strong Rudin's Conjecture: let be N=GP_k+1=> 8 for some integer k, where GP_k is the k-th generalized pentagonal number, then Q(N)=Q(N;q,a) with gcd(q,a) squarefree and q> 0 if and only if (q,a)=(24,1).

math.NT↗

arXiv2013

Markoff-Rosenberger triples in arithmetic progression

We study the solutions of the Rosenberg--Markoff equation ax^2+by^2+cz^2 = dxyz (a generalization of the well--known Markoff equation). We specifically focus on looking for solutions in arithmetic progression that lie in the ring of integers of a number field. With the help of previous work by Alvanos and Poulakis, we give a complete decision algorithm, which allows us to prove finiteness results concerning these particular solutions. Finally, some extensive computations are presented regarding two particular cases: the generalized Markoff equation x^2+y^2+z^2 = dxyz over quadratic fields and the classic Markoff equation x^2+y^2+z^2 = 3xyz over an arbitrary number field.

math.NT↗