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.
Here is an updated program output with different parameters, setting the maxima of A,B,C,D to:
1500 3500 5500 7500
I now believe that requested value of A may depend too heavily on how large the numbers in the search space can be. The present program took about 2.5 hours to run. It seems like there should be an analytic way with one leg of a right triangle known to limit the search to a subset of the possible second leg lengths rather than checking every integer in the range, but I have not thought of any way.
With smaller parameters, there was a 3-way tie for first in the A^2+B^2 contest, but allowing larger numbers broke the tie
ab [840] 21
3330
abc [288] 84
8198
abcd [252] 202
13686
There was a new winner. A=252 squeaked past A=288 for the win. Here are the top ten rankings of A and the number of quadruplets counted.
A count (of {A,B,C,D})
252 202
288 201
504 185
432 182
720 181
576 180
360 168
144 160
180 156
756 155
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Posted by Larry
on 2025-04-17 14:02:41 |
Different Parameters, Different Solution
