## Development of |

The sieve of Helmes uses the periodical appearance of the prime numbers on the irreducible polynoms

1. f(x)=x^2+1 all primes mod 4 = 1 appear as factor

2. f(x)=x^2+x+1 all primes mod 6 = 1 appear as factor

All primes without the prime number 2 appear as factor on these three polynoms

I. f(x)=2x^2-1 all primes mod 8 = 1 or 7 appear as factor

II. g(x)=2x^2+1 all primes mod 8 = 1 or 3 appear as factor

III. h(x)=4x^2+1 all primes mod 8 = 1 or 5 appear as factor

Then there is some x element of N and some {f, g, h} such that p divides t(x)

Assume first that p = 4k+1 for some k element N

Then -1 is a square in Fp* and hence there is some a element Fp* such that a^2= -1.

Put x´:=2^(-1)a element Fp and choose some x element N such that x´= x mod p

then 4x^2 = 4x´^2 = 4*2^(-2)a^2 = a^2 = -1 mod p.

Hence p divides h (x) = 4x^2+1

The remaining primes are of the form 8k+3 or 8k-1.

Assume that p = 8k+3 for some k element N Then -2 is a square in Fp and so is -1/2.

So there is x element N such that x^2 = -1/2 mod p

Hence p divides g(x) = 2x^2+1.

Assume finally that p = 8k-1

Then 2 is a square in Fp and so is 1/2.

So there is x element N such that x^2 = 1/2 mod p

Hence p divides f(x) = 2x^2-1.