--- In

primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:

>

> I recommend

> http://www.alpertron.com.ar/QUAD.HTM

> for blow-by-blow solutions of such problems,

> using continued fractions.

>

The presentation was beautifully done, but the

calculations looked very difficult. A simpler

approach (at least conceptually) should work for

those integer binary quadratic forms where the

number of times a particular value appears is

coordinated perfectly with the number of factors

it has.

For my test number I chose x = 44398336 and

y = 258399491 yielding A = 116050602938281481

as a value of A = 25x^2 + y^2, with the goal of

demonstrating its primality or finding factors. Since 'A'

values that have in common modular residues that match

the ones that the test number posses can only reside on

certain rows of y, all of the non-solutions can be eliminated

with a relatively few moduli. In this case, the twenty-nine

prime moduli listed below sieved out all but the two

locations where the test number actually resides. A

valid x = 68132401 is found at y = 35884 yielding factors

from the formula of 4045597 and 28685655773.

A mod 10000 = 1481, A mod 3 = 2, A mod 7 = 1, A mod 11= 8,

A mod 13= 2, A mod 17= 5, A mod 19= 8, A mod 23= 22,

A mod 29= 28, A mod 31= 14, A mod 37= 26, A mod 41= 9,

A mod 43= 6, A mod 47= 39,A mod 53= 9, A mod 59= 2,

A mod 61= 16, A mod 67= 3, A mod 71= 49, A mod 73= 52,

A mod 79= 18, A mod 83= 58, A mod 89= 26, A mod 97= 54,

A mod 101= 27, A mod 103= 73, A mod 107= 59,

A mod 157= 97, A mod 211= 81

Aldrich