I have formulated the following conjecture:

Odd positive integer $ N=6n-1$ is a prime number iff neither of two diophantine equations

$6x^2+(6x−1)y=n$
 
$6x^2+(6x+1)y=n$

has solution. $x=1,2,3,..y=0,1,2,...n =1,2, 3..$

Odd positive integer $ N=6n+1$ is a prime number iff neither of two diophantine equations

$6x^2−2x+(6x−1)y=n$
 
$6x^2+2x+(6x+1)y=n$

has solution. $x=1,2,3,..y=0,1,2,...n =1,2, 3..$

Note: all primes, except 2 and 3, belong to two sequences:  $S1(n)=6n-1$ or $S2(n)=6n+1$

Can anybody help to prove this statement?