I have derived the following theorem:
Odd positive integer $N=6n-1$ is a prime iff neither of two diophantine equations
$6x^2+(6x-1)y=n$
$6x^2+(6x+1)y=n$
has solution.
Odd positive integer $N=6n+1$ is a prime iff neither of two diophantine equations
$6x^2-2x+(6x-1)y=n$
$6x^2+2x+(6x+1)y=n$
$x=1, 2, 3,..; y=0, 1, 2,...; n=1, 2, 3,...$
has solution.
Theorem allows to find all primes (up to a given limit) since all primes (exept 2 and 3) are in one of two forms $6n-1$ or $6n+1$.
My question is: does exist a general method to determine that a given diophantine equation has
no solution?