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Yes, the upper bound is crucial in finding all the Pythagorean triplets within the time limit.
Note that the M in my upper bound definition is not the same as the M in the problem statement. Sorry for the confusion about that. This might explain the difference in your findings about the upper bound.
The other insight you mention is crucial as well. I didn't want to spoil too much :-)
Project Euler #86: Cuboid route
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Yes, the upper bound is crucial in finding all the Pythagorean triplets within the time limit.
Note that the M in my upper bound definition is not the same as the M in the problem statement. Sorry for the confusion about that. This might explain the difference in your findings about the upper bound.
The other insight you mention is crucial as well. I didn't want to spoil too much :-)
I hope this clears some things up.