I'm working on some basic number theory. I came across an idea and I'm having trouble finding a general solution. A == X mod (L-k) k^2 == Y mod (L-k) It is the equivalent of: ( k^2 - A ) / ( L - k ) = integer A and L are two different pre-chosen numbers. I know L-k is a prime number, I just don't know which prime number from L-1 to 2. Is there a way to find a value for k such that X=Y? An example is A=129 L=130. I used my calculator to find k=99. A=1613 L=1940 k=333 I also used the quadratic formula to find a lower bound [ 4L - sqrt ( (4L)^2 - 4( 2(L^2) + A ) ) ] / 2 I have found the greater: ( k^2 – A ) / ( L – k ) + 2k Becomes, the less accurate the lower bound is. I have tried to apply solving systems of congruences but the method was for linear equations. And the changing modulus made it even harder. Also, when you derive an equation from a previously proven equation, do you have to prove the new equation? I have a copy of Gareth Jones and Mary Jones “Elementary Number Theory” and George Andrews “Number Theory.” Maybe you can point me to something I missed.