Number Theory

0907 Submissions

[2] viXra:0907.0034 [pdf] replaced on 2017-01-28 07:40:47

Essai Sur L'Hypothèse de Riemann

Authors: M. MADANI Bouabdallah
Comments: 8 Pages.

J.P. Gram (1903) writes p.298 in 'Note sur les zéros de la fonction zéta de Riemann' : 'Mais le résultat le plus intéressant qu'ait donné ce calcul consiste en ce qu'il révèle l'irrégularité qui se trouve dans la série des α. Il est très probable que ces racines sont liées intimement aux nombres premiers. La recherche de cette dépendance, c'est-à-dire la manière dont une α donnée est exprimée au moyen des nombres premiers sera l'objet d'études ultérieures.' Also the proof of the Riemann hypothesis is based on the definition of an application between the set P of the prime numbers and the set S of the zeros of ζ.
Category: Number Theory

[1] viXra:0907.0024 [pdf] submitted on 20 Jul 2009

Adjugates of Diophantine Quadruples

Authors: Philip Gibbs
Comments: 7 pages. Published in INTEGERS 10 (2010), 201-209, (The Electronic Journal of Combinatorial Number Theory)

Diophantine m-tuples with property D(n), for n an integer, are sets of m positive integers such that the product of any two of them plus n is a square. Triples and quadruples with this property can be classed as regular or irregular according to whether they satisfy certain polynomial identities. Given any such m-tuple, a symmetric integer matrix can be formed with the elements of the set placed in the diagonal and with corresponding roots off-diagonal. In the case of quadruples, Jacobi's theorem for the minors of the adjugate matrix can be used to show that up to eight new Diophantine quadruples can be formed from the adjugate matrices with various combinations of signs for the roots. We call these adjugate quadruples.
Category: Number Theory