Harristotelian Logic

Consider

(jk - Tk + T)(jk + Tk + T) = T^4

where T = M + 1, or T = M - 1, where M is some integer to be factored.

Solving for k gives

k = (-jT +/- T^2 sqrt(j^2 - T^2 + 1))/(j^2 - T^2)

and the choice for T means that you can factor T^4 so that you have

jk - Tk + T = f_1 and

jk + Tk + T = f_2, where

f_1 f_2 = T^4,

and solve for j, and you'll get rational j's that are not integers for the interesting solutions.

Using those j's you should be able to factor M rather easily, as then you have

k = (-j(M+1) +/- (M+1)^2 sqrt(j^2 - M(M+2))/(j^2 - (M+1)^2)

using T = M+1.

I want readers to consider the insulting posts that came in response to my previous posting, as I want you to think about those people.