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November 17th, 2018, 05:53 PM   #1
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Pythagorean n-tuples' tendency

The following Pythagorean n-tuples determine the set of every value for the exclusive sums of n terms in Pythagorean equations:

For instance,

One-tuples: x^2; 1, 4, 9, 16, 25, 36...

Two-tuples: x^2+y^2; 2, 5, 8, 10, 13, 18...

Three-tuples: x^2+y^2+ z^2; 3, 6, 9, 11, 12, 14...

etc.

Do these Pythagorean n-tuples' frequencies together tend to converge or diverge toward infinity, or otherwise have a finite mode, either minimum or maximum?

Last edited by Loren; November 17th, 2018 at 06:00 PM. Reason: To change one word
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November 18th, 2018, 03:53 AM   #2
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For n-tuples, the first number listed is n. What do you mean by "frequencies"?
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November 18th, 2018, 09:21 AM   #3
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The "frequency" here is the mode (minimum or maximum, for finite frequency) or cardinality (for transfinite frequency) of how often a sum appears as n approaches infinity.

E.g., in the incomplete list below, 9 appears twice. Perhaps Cantor's diagonalization argument could be applied to the complete list.
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