Three positive integers P, Q and R, with P < Q < R, are in arithmetic sequence satisfying :
N*P*Q*R + Q² = R² - P², 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.

NPQR + Q² = R² - P²  gives          NP = (R² - P² - Q²)/(QR)

Writing  R = 2Q - P  and simplifying then gives      NP = (3Q - 4P)/(2Q - P)

which can be rearranged to                                 NP = 3/2  -  5P/(4Q - 2P)
 
Since 4Q > 2P, it follows that NP < 3/2 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, 3, 5, 1 respectively.

Posted by Harry on 2011-07-05 19:15:11