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?
(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! :)
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Posted by tomarken
on 2006-04-26 12:21:18 |
re: Adding to the solution...even numbers!
