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

values only about 3 in 10000 or about 105299 altogether

potentially need to be checked to see if there is a

corresponding (5x)^2 that will generate the chosen 'A'

(or values near it) and factor it. If the entire range

of y is traversed without finding another x value that

gives us 'A' then 'A' is prime. For the given test

number a valid x = 68132401 is found right away at

y = 35884 yielding factors from the formula of 4045597

and 28685655773.

Clearly the number of possible y values that need to be considered could be greatly reduced simply by increasing the number of m values used to make exclusionary sieves, but this would also increase computational overhead, and may turn out to be too expensive. It is not clear if even an optimized version would have the potential to be tooled up to analyze really large integers.