This paper deals with the Diophantine equation axy+bx+cy=d (and bc are coprime). It revisits the old methods of solving this equation. It establishes a link between this solution and the characterization of the elements of the arithmetic progression ax+b (and b are coprime). It provides a new method for solving this equation. It leads to two primality criteria and a commutative diagram characterizing odd natural numbers.
This paper presents, in algebraic form, the set of prime numbers as obtained by the sieve of Eratosthenes and as contained in an arithmetic progression. In this way, it unifies old and recent studies on prime numbers: Euclid’s theorem, Dirichlet’s theorem, Green-Tao’s theorem, the conjecture of twin primes (generalized by Polignac) and Chebyshev’s Bias Phenomenon. It re-demonstrates the three theorems, solves Chebyshev’s Bias Phenomenon, demonstrates the twin primes conjecture and elucidates Polignac’s conjecture.
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