From the Diophantine equation ab − (a + b + c) = k2 − 1 to LMFDB Curve 1576.a1 and a Degree-Two del Pezzo Surface

This preprint/report explores the Diophantine condition where twice the area minus the perimeter of a right-angled triangle is one less than a perfect square (2K - U = k² - 1). What begins as an elementary numerical observation on Pythagorean triples naturally unfolds into a rigorous study of algebraic surfaces!

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-08-26
DOI
https://doi.org/10.5281/zenodo.22107951
Primary Topic
Algebraic Geometry and Number Theory
Type
preprint
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preprint

From the Diophantine equation ab − (a + b + c) = k2 − 1 to LMFDB Curve 1576.a1 and a Degree-Two del Pezzo Surface

Pascal T. Grülling
Zenodo (CERN European Organization for Nuclear Research)
Algebraic Geometry and Number Theory
preprint

From the Diophantine equation ab − (a + b + c) = k2 − 1 to LMFDB Curve 1576.a1 and a Degree-Two del Pezzo Surface

Pascal T. Grülling
preprint en

Abstract

This preprint/report explores the Diophantine condition where twice the area minus the perimeter of a right-angled triangle is one less than a perfect square (2K - U = k² - 1). What begins as an elementary numerical observation on Pythagorean triples naturally unfolds into a rigorous study of algebraic surfaces!

Zenodo (CERN European Organization for Nuclear Research)
Algebraic Geometry and Number Theory
DOIOpen Access PDFOpenAlex
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From the Diophantine equation ab − (a + b + c) = k2 − 1 to LMFDB Curve 1576.a1 and a Degree-Two del Pezzo Surface — Pascal T. Grülling · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS