There are infinitely many sets of positive integers A<B<C<D such that A^2+B^2, A^2+B^2+C^2, and A^2+B^2+C^2+D^2 are all squares. Find the value of A≤ 1500 which leads to the largest number of solutions.
Starting with the smaller problem of just A^2+B^2, and looking at the number of solutions for different values of A, it appears that A values with more solutions tend to have as many prime factors as possible. (I am not sure why this should be)
For example, some A values with a larger number of solutions (to A^2+B^2 is a square), and their prime factorizations, followed by number of solutions when checking B values up to a certain maximum:
840 [2, 2, 2, 3, 5, 7] 35
1260 [2, 2, 3, 3, 5, 7] 30
1320 [2, 2, 2, 3, 5, 11] 30
Of course, the higher the upper limit, the more solutions are found and the longer the run times.
For this preliminary look, 840 seems promising.
Checking further, for A^2+B^2+C^2, here are some key values
A number (max B and C = 10000)
840 96
1260 98
1320 59
A number (max B and C = 12000)
840 116
1260 115
1320 75
So 840 or 1260 are possibilities, pending further evaluation. Better than a guess, but not at all definitive.
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Posted by Larry
on 2025-04-14 19:26:52 |
Preliminary investigations
