Diophantine $m$-tuples and elliptic curves
نویسندگان
چکیده
منابع مشابه
Irregular Diophantine m-tuples and elliptic curves of high rank
A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them is one less than a perfect square. In this paper we characterize the notions of regular Diophantine quadruples and quintuples, introduced by Gibbs, by means of elliptic curves. Motivated by these characterizations, we find examples of elliptic curves over Q with torsion group Z/2Z × Z/2Z and ...
متن کاملDiophantine m-tuples for primes
In this paper, we show that if p is a prime and ifA = {a1, a2, . . . , am} is a set of positive integers with the property that aiaj +p is a perfect square for all 1 ≤ i < j ≤ m, then m < 3 · 2168. More generally, when p is replaced by a squarefree integer n, the inequality m ≤ f(ω(n)) holds with some function f , where ω(n) is the number of prime divisors of n. We also give upper bounds for m ...
متن کاملDiophantine m-tuples for linear polynomials
In this paper, we prove that there does not exist a set with more than 26 polynomials with integer coefficients, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients. 1991 Mathematics Subject Classification: 11D09.
متن کاملOn the size of Diophantine m-tuples
Let n be a nonzero integer. A set of m positive integers {a1, a2, . . . , am} is said to have the property D(n) if aiaj + n is a perfect square for all 1 ≤ i < j ≤ m. Such a set is called a Diophantine m-tuple (with the property D(n)), or Pn-set of size m. Diophantus found the quadruple {1, 33, 68, 105} with the property D(256). The first Diophantine quadruple with the property D(1), the set {1...
متن کاملDiophantine m-tuples for quadratic polynomials
In this paper, we prove that there does not exist a set with more than 98 nonzero polynomials in Z[X], such that the product of any two of them plus a quadratic polynomial n is a square of a polynomial from Z[X] (we exclude the possibility that all elements of such set are constant multiples of a linear polynomial p ∈ Z[X] such that p2|n). Specially, we prove that if such a set contains only po...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal de Théorie des Nombres de Bordeaux
سال: 2001
ISSN: 1246-7405
DOI: 10.5802/jtnb.308