0021e: Article 15 (A Quartic Diophantine Equation)
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 xi, yi > 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,
x1k+nx2k = y1k+ny2k
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.