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 How many perfect squares (Posted on 2006-04-25)
Suppose a and b are positive integers. We all know that aČ+2ab+bČ is a perfect square. Give an example where also aČ+ab+bČ is a perfect square. How many such examples exist?

 See The Solution Submitted by Salil Rating: 3.0000 (2 votes)

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 re(2): Partial Solution | Comment 3 of 19 |
(In reply to re: Partial Solution by Charlie)

Here's just the beginning of what seems a never-ending list of the beginnings of families of solutions. The only member of the family shown is the one in which the gcd of the two numbers is 1.

`3       57       85       1611      247       3313      3516      399       5632      4517      6340      5111      8519      8055      5740      7724      9513      12023      12065      8869      9125      14356      11515      16175      11232      175104     10556      16529      19517      20887      16085      16831      224119     14572      20319      26193      18764      221144     15595      21740      279`
`  10   T=3  20   while (Ct<40)  30    for I=1 to int(T/2)  40     J=T-I  50     if gcd(I,J)=1 then  60       :V=I*I+I*J+J*J  70       :Sr=int(sqrt(V)+0.5)  80       :if Sr*Sr=V then print I,J:Ct=Ct+1:endif  90    next  95    T=T+1 100   wend`

 Posted by Charlie on 2006-04-25 10:05:05
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