[seqfan] Re: Primes of the form 8n + 7 or -x + 4xy + 4y^2

Don Reble djr at nk.ca
Mon Nov 14 08:54:02 CET 2016

Alonzo and all:

> ... does not guarantee that for every n there are x and y that
> will give the same value as 8n + 7 in -x^2 + 4xy + 4y^2.
> So that's what I'm going to be thinking about now.

    The Quadratic Form and Quadratic Reciprocity theories cover that.

    Z = (-x^2 + 4xy + 4y^2) is a primitive quadratic form with
    discriminant 32. The QF theory shows that there are essentially
    two such forms, A = (8u^2 - v^2) and B = (u^2 - 8v^2).
        (Substitute x->2u+v, y->u to show that Z is like A.)
    Every primitive, discr=32 form is like A or B: that is,
    the range of each such form is the same as A's range or B's range.

    If P is a 8n+7 prime, then 2 is a quadratic residue (QR theory):
    there exists a T, 0<T<P, such that T^2 == 2 mod P.
    P divides T^2 - 2, and U = (16T^2 - 32)/4P is an integer.

    The form C = (Px^2 + 4Txy + Uy^2) has discriminant 32, is primitive,
        (T < P and P is an odd prime; therefore gcd(P,4T)=1)
    and represents P (at x=1 y=0).
    C isn't like form B (B doesn't represent any 8n+7 numbers)
    so it is like form A, and A represents P.

    8n+1 primes also have quadratic residue 2, so analogously
    form B represents all of those primes.  A007519 = A141174.

Don Reble  djr at nk.ca

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