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Figuring Forty-Nine and Sequence (Posted on 2016-10-25) Difficulty: 3 of 5
Find all possible triples (P, Q, R) of positive integers satisfying both of the following conditions

49, P2 and Q2 (in order) are in arithmetic sequence with common difference > 0 , and:

49, Q2 and R2 (in order) are in arithmetic sequence with common difference > 0

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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No Subject Comment 2 of 2 |
From the first AP, 2p^2=q^2 +49, and from the second, 2q^2=r^2 +49.

Eliminate q to get 4p^2 - r^2 = 147 = 3*7*7.

Factor LHS and solve to find p=(37,13,7).

The first solution doesn't lead to rational q and the third leads to AP with zero difference between terms.

From the second (p,q,r)=(13,17,23) matching the output from Charlie's program.

  Posted by xdog on 2016-10-25 12:05:03
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