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An Analytical Study of Diophantine Equations of Pythagorean Form: Causal Inferences on Hypothesized Relations between Quadratic and Non-Quadratic Triples

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Title: An Analytical Study of Diophantine Equations of Pythagorean Form: Causal Inferences on Hypothesized Relations between Quadratic and Non-Quadratic Triples
Language: English
Authors: Carmelo R. Cartiere
Source: Athens Journal of Education. 2025 12(3):527-546.
Availability: Athens Institute for Education & Research. 8 Valaoritou Street, Kolonaki, Athens 10671, Greece. e-mail: education@atiner.gr; Web site: https://www.athensjournals.gr/aje
Peer Reviewed: Y
Page Count: 20
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Descriptors: Mathematics Education, Calculus, Validity, Mathematical Logic, History, Mathematics, Equations (Mathematics), Problem Solving
ISSN: 2407-9898
2241-7958
Abstract: In XVII century, presumably between 1637 and 1638, with a note in the margin of Diophantus' "Arithmetica", Pierre de Fermat stated that Diophantine equations of the Pythagorean form, x[superscript n] + y[superscript n] = z[superscript n], have no integer solutions for n > 2, and (x, y, z) > 0. Of this statement, however, Fermat never provided a proof. Only after more than 350 years, in 1994, Prof. Andrew J. Wiles was finally successful in demonstrating it (Wiles, 1995; Taylor & Wiles, 1995; Boston, 2008). However, Wiles' proof adopts calculus techniques far beyond Fermat's knowledge. Our aim is to show an analytical method to attempt a proof to Fermat's last theorem with the only use of elementary calculus techniques.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1480305
Database: ERIC
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  Availability: 0
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  Data: Athens Institute for Education & Research. 8 Valaoritou Street, Kolonaki, Athens 10671, Greece. e-mail: education@atiner.gr; Web site: https://www.athensjournals.gr/aje
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  Data: In XVII century, presumably between 1637 and 1638, with a note in the margin of Diophantus' "Arithmetica", Pierre de Fermat stated that Diophantine equations of the Pythagorean form, x[superscript n] + y[superscript n] = z[superscript n], have no integer solutions for n > 2, and (x, y, z) > 0. Of this statement, however, Fermat never provided a proof. Only after more than 350 years, in 1994, Prof. Andrew J. Wiles was finally successful in demonstrating it (Wiles, 1995; Taylor & Wiles, 1995; Boston, 2008). However, Wiles' proof adopts calculus techniques far beyond Fermat's knowledge. Our aim is to show an analytical method to attempt a proof to Fermat's last theorem with the only use of elementary calculus techniques.
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  Data: 2025
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  Data: EJ1480305
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