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How many perfect squares (Posted on 2006-04-25) Difficulty: 3 of 5
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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Solution re: Adding to the solution...even numbers! | Comment 12 of 19 |
(In reply to Adding to the solution... by tomarken)

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  39
24  95
32  45
32  175
36  135
40  51
40  77
40  279
48  117
48  407
56  115
56  165
56  559
64  221
64  735
72  203
72  285
80  195
80  357
88  315
88  437
96  525
104  105
104  451
112  273
120  153
120  231
136  209
144  155
144  351
152  273
160  225
168  345
176  259
176  429
184  425
200  255
200  385
208  387
224  315
240  253
240  287
264  325
280  357
280  423
288  405
312  315
360  459
432  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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