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A note on the number of S -Diophantine quadruples

Florian Luca, Volker Ziegler (2014)

Communications in Mathematics

Let ( a 1 , , a m ) be an m -tuple of positive, pairwise distinct integers. If for all 1 i < j m the prime divisors of a i a j + 1 come from the same fixed set S , then we call the m -tuple S -Diophantine. In this note we estimate the number of S -Diophantine quadruples in terms of | S | = r .

Class Number Two for Real Quadratic Fields of Richaud-Degert Type

Mollin, R. A. (2009)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary: 11D09, 11A55, 11C08, 11R11, 11R29; Secondary: 11R65, 11S40; 11R09.This paper contains proofs of conjectures made in [16] on class number 2 and what this author has dubbed the Euler-Rabinowitsch polynomial for real quadratic fields. As well, we complete the list of Richaud-Degert types given in [16] and show how the behaviour of the Euler-Rabinowitsch polynomials and certain continued fraction expansions come into play in the complete determination...

La conjecture de Dickson et classes particulières d’entiers

Abdelmadjid Boudaoud (2006)

Annales mathématiques Blaise Pascal

En admettant la conjecture de Dickson, nous démontrons que, pour chaque couple d’entiers q &gt; 0 et k &gt; 0 , il existe une partie infinie L q , k telle que, pour chacun des entiers n L q , k et tout entier s tel que 0 &lt; s q , on ait n + s = s t 1 . . . t k t 1 &lt; . . . &lt; t k sont des nombres premiers. De même, pour chaque couple d’entiers q &gt; 0 et k &gt; 0 , il existe une partie infinie M q , k telle que, pour chacun des entiers n M q , k et tout entier s (nul ou non ) de l’intervalle - q , q , on ait n + s = l t 1 . . . t k t 1 &lt; . . . &lt; t k sont des nombres premiers et l’entier l appartient à l’intervalle 1 , 2 q + 1 . La lecture non standard...

Modifications of the Eratosthenes sieve

Jerzy Browkin, Hui-Qin Cao (2014)

Colloquium Mathematicae

We discuss some cancellation algorithms such that the first non-cancelled number is a prime number p or a number of some specific type. We investigate which numbers in the interval (p,2p) are non-cancelled.

On polynomials that are sums of two cubes.

Christopher Hooley (2007)

Revista Matemática Complutense

It is proved that, if F(x) be a cubic polynomial with integral coefficients having the property that F(n) is equal to a sum of two positive integral cubes for all sufficiently large integers n, then F(x) is identically the sum of two cubes of linear polynomials with integer coefficients that are positive for sufficiently large x. A similar result is proved in the case where F(n) is merely assumed to be a sum of two integral cubes of either sign. It is deduced that analogous propositions are true...

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