## The answer key for the Prize Puzzle \$\$ of 8/29/10.

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• It concerns A = nx^2 + nxy + y^2 for n = 2,3,5,6,8, and 14, and the unique issquare function fr^2 - gA = hS^2 associated with each of the binary quadratic
Message 1 of 1 , Sep 14, 2010
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It concerns A = nx^2 + nxy + y^2
for n = 2,3,5,6,8, 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,x1,y1,x2,y2,
r1,r2,r3,r4,s1,s2,s3,s4,j1,j2 : integers;

At n = 2, A = 2x^2 + 2xy + y^2 and its
applicable issquare function is -r^2 + 2A = s^2.

At n = 3, A = 3x^2 + 3xy + y^2 and its
applicable issquare function is -3r^2 + 4A = s^2.

At n = 5, A = 5x^2 + 5xy + y^2 and its
applicable issquare function is 5r^2 - 4A = s^2.

At n = 6, A = 6x^2 + 6xy + y^2 and its
applicable issquare function is 3r^2 - 2A = s^2.

At n = 8, A = 8x^2 + 8xy + y^2 and its
applicable issquare function is 2r^2 -A = s^2.

At n = 14, A = 14x^2 + 14xy + y^2 and its
applicable issquare function is 7r^2 - 2A = 5s^2.

Example at n = 14:
Two values each for r and s, namely r1,r2,s1,s2, may
be obtained directly from the x,y coordinates (x1,y1)
of any A by means of simple formulas:
r1 = 2x1 + y1,
r2 = r1 + 10x1,
s1 = y1,
s2 = r1 + r2 -s1
If an identical A value exists with different x,y
coordinates, then values for r3,r4,s3,and s4 may
found in the same way, and two factors j1,j2 of A
may be easily calculated:
j1,j2 = GCD(A, (r1 + r2)*r3 +/- (r2 - r1)*s3).
If A = 40889 then x1 =32, y1 =53, x2 =37, y2 =39
and the two factors are 31 and 1319.

Where have all the prize hawks gone,long time passing?
Doesn't anyone believe in a free lunch anymore?
It is another gray day in Dog Patch.

Aldrich Stevens
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