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 $S2(n)=6n+1$ Theorem allows to substitute the task: "Find all primes in the range $(N_1;N_2)$" by the task: "Find positive integers which do not appear in the range $(n_1;n_2)$ in two pairs of $2$-dimensional arrays
$P_1(i,j)=6i^2+(6i-1)(j-1)$
$P_2(i,j)=6i^2+(6i+1)(j-1)$
$i,j = 1,2,3,\ldots$
for primes in the sequence $N=6n-1$.
$P_3(i,j)=6i^2-2i+(6i-1)(j-1)$
$P_4(i,j)=6i^2+2i+(6i+1)(j-1)$
$i,j = 1,2,3,\ldots$
for primes in the sequence $N=6n+1$. Since all primes (except $2$ and $3$) are in one of two forms $6n−1$ or $6n+1$.
Can anybody help to prove this statementadvice me - is proposed "matrix sieve" algorithm well-known?
