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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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re(4): Partial Solution | Comment 7 of 19 |
(In reply to re(3): Partial Solution by Charlie)

With regard to base pairs that include a prime, each prime (>=5) seems to be a member of two such base pairs:

5       3
5       16
7       8
7       33
11      24
11      85
13      35
13      120
17      63
17      208
19      80
19      261
23      120
23      385
29      195
29      616
31      224
31      705
37      323
37      1008
41      399
41      1240
43      440
43      1365
47      528
47      1633
53      675
53      2080
59      840
59      2581
61      899
61      2760
67      1088
67      3333
71      1224
71      3745
73      1295
73      3960
79      1520
79      4641

I don't know if composite numbers have a similar theme.


  Posted by Charlie on 2006-04-25 11:16:22
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