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11.
Bérczes Attila Dujella Andrej Hajdu Lajos Tengely Szabolcs 《Monatshefte für Mathematik》2016,180(3):469-484
Monatshefte für Mathematik - Diophantine sets, i.e. sets of positive integers A with the property that the product of any two distinct elements of A increased by 1 is a perfect square, have a... 相似文献
12.
We describe a decomposition of the polynomials f
n
:= f
n,B,a
defined by
where B and a are rational numbers. We also present an application to related Diophantine equations.
The authors were supported by the Ministry of Science, Education and Sports, Republic of Croatia, grant 0037110. 相似文献
13.
Julián Aguirre Andrej Dujella Mirela Jukić Bokun Juan Carlos Peral 《Periodica Mathematica Hungarica》2014,68(2):222-230
There are 26 possibilities for the torsion groups of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with a given torsion group which set the current rank records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for \(\mathbb {Z}/15\mathbb {Z}\) , there exists an elliptic curve over some quadratic field with this torsion group and with rank \(\ge 2\) . 相似文献
14.
In this paper, we prove that if {a,b,c,d} is a set of four non-zero polynomials with integer coefficients, not all constant, such that the product of any two of its distinct elements plus 1 is a square of a polynomial with integer coefficients, then
(a+b−c−d)2=4(ab+1)(cd+1). 相似文献
15.
Archiv der Mathematik - In this paper, we give a nontrivial lower bound for the fundamental unit of norm $$-1$$ of a real quadratic field of class number 1. 相似文献
16.
We consider arithmetic progressions consisting of integers which are y-components of solutions of an equation of the form x 2 ? dy 2 = m. We show that for almost all four-term arithmetic progressions such an equation exists. We construct a seven-term arithmetic progression with the given property, and also several five-term arithmetic progressions which satisfy two different equations of the given form. These results are obtained by studying the properties of a parametric family of elliptic curves. 相似文献
17.
Alon, Angel, Benjamini and Lubetzky [1] recently studied an old problem of Euler on sumsets for which all elements of A+B are integer squares. Improving their result we prove: 1. There exists a set A of 3 positive integers and a corresponding set B?[0,N] with |B|?(logN)15/17, such that all elements of A+B are perfect squares. 2. There exists a set A of 3 integers and a corresponding set B?[0,N] with |B|?(logN)9/11, such that all elements of the sets A, B and A+B are perfect squares. The proofs make use of suitably constructed elliptic curves of high rank. 相似文献