## Speculation concerning the primes and composites of A = 25x^2 + y^2.

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• {A,x,y,F,m : positive integers} Consider for a moment the values of A that are +/- 1 mod 10. Exclude all values of A with Gcd(A,y) 1, as well as all
Message 1 of 1 , Aug 4, 2012
{A,x,y,F,m : positive integers}

Consider for a moment the values of 'A' that are
+/- 1 mod 10. Exclude all values of 'A' with
Gcd(A,y) > 1, as well as all values that are powers
of a prime. If one of the remaining 'A' values appears
more than once it is composite, but if it appears once
only then it is prime, and all of them will be composed
solely of +1 mod 4 primes. Factors for the composite
terms may be found by putting the values of two of their
(x,y) pairs into the formula
F = Gcd (A, (x2-x1)*(y1+y2) + (x2 + x1)*(y1-y2)).

In my number theory books chaos generally reigns among
the patterns of primes and composites produced by the
binary quadratic forms cited there and most of the time
there is no correlation between the number of factors
of a term and the number of times it appears as a value.
I could find no mention of rare forms such as
A = 25x^2 + y^2, A = 1320x^2 + y^2, A = 5x^2 + 5xy + y^2,
A = 2x^2 + 2xy + y^2, etc. that make primes, composites
and excluded values easily identifiable in well ordered
equations, even though the last two of these together have
as values three quarters of the entire set of primes!
(+3 mod 4 primes that are also +/- 3 mod 10 are not covered).Certainly it seems possible that the kind of
symmetry exhibited by these structures may turn out to be
useful and there is still some hope that an obscure
chapter or two on the topic may reside somewhere in the
archives. However neither Hardy/Wright nor Crandall/
Pomerance identify them as a category unto themselves.

One obvious strategy for exploiting their characteristics
is simply to develop a search method to see how many times
a desired value or a batch of values occurs among all the
'A' values of a similar size. The difficulty with this
approach is that the results can be incomplete unless the
entire range of y is traversed. Fortunately, this problem
may be ameliorated somewhat.

For my test number I chose x = 44398336 and y = 258399491
yielding A = 116050602938281481, with the goal of
demonstrating its primality or finding factors. It was
not hard to write a little programming routine to
discover that 'A' values ending in 1481 will be located
somewhere on each of only 88 rows of y repeated in a
cycle of 5000. Of these possibilities many can be
eliminated because for each modulus (m prime) only a bit
more than half of the remaining y will have a term
somewhere that ends in 1481 and also has the same residue
for some A mod m as the test number (here m = 3,7,11,13,17,19,23,31,61,157 and 211). If several arrays
are built identifying which y values may still possess
another copy of the target integer using the chosen
moduli then we find that of about 340000000 original y