[2] **viXra:0907.0034 [pdf]**
*replaced on 2013-02-20 07:32:38*

**Authors:** M. MADANI Bouabdallah

**Comments:** 04 Pages. French language

We try to prove by elementary geometry the Conjecture on Primes Numbers n² + 1.

**Category:** Number Theory

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

**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