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## Main Question or Discussion Point

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.

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.