We consider Diophantine quintuples a, b, c, d, e. These are sets of positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most Diophantine quintuples.
The diophantine equation (1) x³ + y³ + z³ = nxyz has only trivial solutions for three (probably) infinite sets of n-values and some other n-values ([7], Chs. 10, 15, [3], [2]). The main set is characterized by: n²+3n+9 is a prime number, n-3 contains no prime factor ≡ 1 (mod 3) and n ≠ - 1,5. Conversely, equation (1) is known to have non-trivial solutions for infinitely many n-values. These solutions were given either as "1 chains" ([7], Ch. 30, [4], [6]), as recursive...
In this paper we consider some special classes of Diophantine equations connected with McFarland's and Ma's conjectures about difference sets in abelian groups and we obtain an extension of known results.
Soit un entier . Pour et , nous considérons la suite de Lucas . Nous montrons que, pour n’est ni un carré, ni le double, ni le triple d’un carré, ni six fois un carré pour sauf si et .