Perfect Powers in Pythagorean Triples
Year: 2026
Type: software
Abstract: This expository note gives a negative answer to the Mathematics Stack Exchange question “Can a Pythagorean Triple ever have TWO identical exponents > 1?” For every integer k ≥ 2, at most one entry of a positive integer Pythagorean triple can be a perfect kth power of a positive integer. The proof applies to both primitive and nonprimitive triples. It combines the Darmon-Merel theorem, Fermat's descent, and the result of Bennett, Chen, Dahmen and Yazdani for the exponent triple (2, 6, 6). A reduction to coprime bases handles arbitrary triples; the signed-integer form of the Darmon-Merel theorem handles every odd exponent k ≥ 5. As a corollary, the exponents in any representation of the three entries as positive integer powers are pairwise coprime. In particular, two representation exponents cannot be identical and greater than one. An example illustrates that distinct exponents greater than one can occur. The note presents a consequence of established theorems, with references to the original sources and the originating question. ChatGPT assisted with finding references, organizing the argument, and drafting and formatting the note; this assistance is acknowledged in the PDF. The accompanying archive contains the PDF, its LaTeX source, and submission information. ## Suggested record details - Creator: vengy (the public byline used in the PDF). - Resource type: Preprint. - Version: 1.0. - Publication date: 2026-10-08, or the date you publish the record. - Language: English. - Optional keywords: Pythagorean triples; perfect powers; number theory; Diophantine equations; generalized Fermat equations. - License: select the license you want to grant in the Zenodo form. ## Files to upload - Perfect_Powers_in_Pythagorean_Triples.pdf - Perfect_Powers_in_Pythagorean_Triples_Zenodo_v1.0.zip The PDF is available separately for immediate viewing. The archive includes an identical copy of that PDF and the editable source. ## Related question https://math.stackexchange.com/questions/3942274/can-a-pythagorean-triple-ever-have-two-identical-exponents-1 ## References 1. H. Darmon and L. Merel, *Winding quotients and some variants of Fermat's Last Theorem*, Journal für die reine und angewandte Mathematik 490 (1997), 81-100. https://doi.org/10.1515/crll.1997.490.81 2. W. Aitken, *Fermat's proof*, expository notes, September 26, 2010. https://public.csusm.edu/aitken_html/notes/FermatsProof.pdf 3. M. A. Bennett, I. Chen, S. R. Dahmen and S. Yazdani, *Generalized Fermat equations: A miscellany*, International Journal of Number Theory 11 (2015), no. 1, 1-28. https://doi.org/10.1142/S179304211530001X 4. vengy, Question 3942274 on Mathematics Stack Exchange, December 10, 2020. https://math.stackexchange.com/questions/3942274/can-a-pythagorean-triple-ever-have-two-identical-exponents-1
Author: dr vengy
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220999474
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