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Some Quadratics Equal Square (Posted on 2009-08-10) Difficulty: 3 of 5
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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Some Thoughts 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 T
OK
run
 1  1            4  4            2  2
 11  16          169  289        13  17
OK

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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