My Math Forum Pythagorean n-tuples' tendency

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 November 17th, 2018, 05:53 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 424 Thanks: 27 Math Focus: Number theory 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
 November 18th, 2018, 03:53 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,759 Thanks: 2138 For n-tuples, the first number listed is n. What do you mean by "frequencies"?
 November 18th, 2018, 09:21 AM #3 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 424 Thanks: 27 Math Focus: Number theory 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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