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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: Adding to the solution...even numbers! | Comment 12 of 19 |

Wow...there also appear to be infinitely many base pairs where 'a' is an even multiple of 8 (the only even number not divisible by 8 that I found that works is 36).  It seems that most multiples of 8, if not all of them, generate more than one base pair.

Here is a "small" sampling:

`a    b-----------16   3924   9532   4532   17536   13540   5140   7740   27948   11748   40756   11556   16556   55964   22164   73572   20372   28580   19580   35788   31588   43796   525104  105104  451112  273120  153120  231136  209144  155144  351152  273160  225168  345176  259176  429184  425200  255200  385208  387224  315240  253240  287264  325280  357280  423288  405312  315360  459432  465`

Check out the following subset of possible solutions:

`a    b      ----------16   39 24   95 32   175 40   279 48   407 56   559 64   735 72   935 80   1159 88   1407 96   1679 104  1975 112  2295 120  2639 128  3007 etc.`
`The b values in this subset can be found by:`

b = ((3a^2 - 8a)/16) - 1

where a is a multiple of 8 (note that a = 8 generates the pair 7 & 8, which was found previously using the odd number formula).

I am pretty fascinated by this problem for some reason, so I will probably continue my search for more equations...

BTW, I should really learn a computer language or something - it took me over an hour to find and compile information by hand using Excel that Charlie probably could have printed out in a matter of seconds! :)

 Posted by tomarken on 2006-04-26 12:21:18

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