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 Some Quadratics Equal Square (Posted on 2009-08-10)
Determine all possible pair(s) (P, Q) of positive integers such that each of P2 + 3Q and Q2 + 3P is a perfect square.

 See The Solution Submitted by K Sengupta Rating: 3.0000 (1 votes)

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 computer exploration -- possible solution | Comment 1 of 2

Due to the symmetric relation of P and Q, it is necessary to check only the cases where P<=Q, and then swap P and Q for any solutions found.

`list   10   for T=1 to 10000   20     for P=1 to int(T/2)   30        Q=T-P   40        V1=P*P+3*Q:V2=Q*Q+3*P   50        Sr1=int(sqrt(V1)+0.5)   60        Sr2=int(sqrt(V2)+0.5)   70        if Sr1*Sr1=V1 then   80         :if Sr2*Sr2=V2 then   90          :print P;Q,V1;V2,Sr1;Sr2  100     next P  110   next TOKrun 1  1            4  4            2  2 11  16          169  289        13  17OK`

indicating that among all cases where P+Q <= 10,000, there are three solutions: (1,1), (11,16) and (16,11).

The squares involved in the first case are both 2^2 = 4.

The squares involved in the latter two cases are 13^2=169 and 17^2=289.

 Posted by Charlie on 2009-08-10 12:49:50

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