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