## 24379Re: Impossible to Prove ??

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• Aug 11, 2012
>
> I recommend
> 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
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