JSH: Bait
- From: Enrico <ungernerik@xxxxxxx>
- Date: Sat, 14 Feb 2009 12:35:55 -0800 (PST)
D is an odd composite to be factored.
Method 1:
=======
D = S^2 + R
(S*X + Y)^2 - D*X^2 = -R*X^2 +2*S*X*Y + Y^2
Find X, Y to produce a square on the right.
Method 2:
=======
D = S^2 + R
Define an easily factored surrogate T(A, B, C)
(B^2 + R) / 2 = A * C
with A an odd number
Note that:
(A + B)^2 + R = A*(A + 2*B + 2*C)
Define another surrogate U(X, Y, Z)
(Y^2 -1) / 2 = X * Z
with X odd and Z a multiple of 4
Note that:
(X + Y)^2 -1 = X*(X + 2*Y + 2*Z)
Define S = T dot U = A*X + B*Y + C*Z
Then
(S*X + C)^2 - D*X^2 = (Y*C + X*B)^2
and
(S*Z + A)^2 - D*Z^2 = (Y*A + Z*B)^2
Enrico
.
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