What question did this study set out to answer?

To derive parametric equations for integer solutions to A^3 = B^3 + C^3 + D^3 with A = B + 1.

April 13, 2026Open Access

Parametric Equations for Three Series of "Quadruples" for the Formula A³ = B³ + C³ + D³

Key Points

To derive parametric equations for integer solutions to A^3 = B^3 + C^3 + D^3 with A = B + 1.
Developed explicit parametric formulas for three families of solutions.
Presented closed-form expressions for each family.
Listed the first 20 numerical examples for clarity.
Identified three distinct infinite families of integer solutions.
Proved the existence of solutions with A = B + 1.
Enumerated examples to support the derived formulas.

Abstract

This paper is a supplement to the previous work “On the First Series of “Quadruples”of Numbers by Analogy with Pythagorean “Triples” for the Formula A³ = B³ +C³ +D³” (April 8, 2026). Here we present explicit parametric formulas for three infinite familiesof integer solutions to A³ = B³ + C³ + D³ with A = B + 1, which were originally givenin tabular form. For each family we derive closed-form expressions and list the first 20numerical examples.

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Cite This Study

Emma Helmdach (Sat,) studied this question.

synapsesocial.com/papers/69dc88b93afacbeac03ea6e6 https://doi.org/https://doi.org/10.5281/zenodo.19510772

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