Prize Puzzle $$ concerning A = nx^2 + nxy + y^2
- for n = 2,3,6,8, and 14, and the unique issquare
function fr^2 - gA = hS^2 associated with each of
the binary quadratic forms defined by n.
A,x,y,n,r,s,f,g,h : integers;
It is a gray day in Dog Patch. Basically I am
positing that if a given value of A on one of
the above equations occurs only once on that
equation for all (x,y), then that is in itself
sufficient proof of primality; that A is either
prime, a square of a prime, or a prime equal to
A/gcd(A,y^2). At n = 14 the additional condition
of removing any extraneous factors of 5 also
If a given value of A occurs more then once on
one of the above equations then A is composite,
and two factors of A can easily be calculated
from each (x,y) pair.The reason this works is
the close connection of x,y to values of r,s of
issquare. At n = 5, A = 5x^2 + 5xy + y^2
and the applicable issquare function is
5r^2 -4A = s^2. if n= 5 and A = 209 = 11* 19,
then A exists at (1,12) and (5,3) and issquare
determines which simple algebraic formulas are
needed to find the r values 14, 15, 13, and 18
where s is an integer directly from the x,y pairs.
These r and s values can in turn be used to
calculate factors of A (no matter how large).
The binary quadratic forms at n = 2,3,6,8 and 14
each has its own issquare function that will apply
to all of its terms. This function will be similar
but not identical to the function 5r^2 -4A = s^2
at n = 5.
A sum of $100 is offered to the first person
to correctly identify the issquare functions
that will apply to the binary quadratic forms
at n = 2,3,6,8, and 14.