0021e: Article 15 (A Quartic Diophantine Equation)

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On The Quartic Diophantine Equation a⁴+nb⁴ = c⁴+nd⁴

by Tito Piezas III

I. Table

In the 1995 paper, “On the Diophantine Equation A⁴+hB⁴ = C⁴+hD⁴”, Choudhry gave non-trivial solutions for 75 integer n in the range 1≤ n ≤ 101. In the table below, we complete this range. The solutions are arranged such that a < c. (The 26 n in blue have been found by this author.)

Table 1. Solutions to a⁴+nb⁴ = c⁴+nd⁴ for all integer 1≤ n ≤101

Author’s note: It seems the range n < 10² is easily managed by today’s computers, with solutions relatively small and only n = 92 needing terms x_i, y_i > 10³. For the extended range n < 10³, it should be interesting to know which n needs the largest initial values.

II. Open Problem

The equation,

x₁^k+nx₂^k = y₁^k+ny₂^k

has parametrizations for any n for k = 2, 3 as shown by,

(ac+nbd)² + n(bc-ad)² = (ac-nbd)² + n(bc+ad)²

(1-9n)³ + n(9n)³ = 1 + n(-3+9n)³

It then raises the question, already asked by others,

Is a⁴+nb⁴ = c⁴+nd⁴ solvable in non-trivial integers (that is a⁴ ≠ c⁴) for any integer n?

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© Tito Piezas III, Sept 2013

You can email author at tpiezas@gmail.com.

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