Following on from this problem, three positive integers P, Q and R, (from smallest to largest in that order), are in arithmetic sequence satisfying : N*P*Q*R + Q = R^2 - P^2, where N is a positive integer.

Determine all possible quadruplet(s) (P, Q, R, N) that satisfy the above equation, and prove that no other quadruplet satisfies the given conditions.

Note that in this variant, the second term involving Q is not a square.

I believe the product NP is constrained much as in the original problem, so arguing along the same lines...

NPQR + Q = R² - P²  gives           NP = (R² - P² - Q)/(QR)

Writing  R = 2Q - P  and simplifying then gives      NP = (4Q - 4P - 1)/(2Q - P)

which can be rearranged to                                 NP = 2  -  (1 + 2P)/(2Q - P)

Since 2Q > P, it follows that NP < 2 and, since N and P are both positive integers, we have N = P = 1 as the only possibility.

Substituting then gives P, Q, R, N  to be 1, 2, 3, 1 respectively.

Posted by Harry on 2011-07-25 19:11:41