The Diophantine equation 𝑥⁴+2𝑦⁴=𝑧⁴+4𝑤⁴

Abstract

Abstract. We show that, within the hypercube |x|, |y|, |z|, |w| ≤ 2.5 · 10 6 , the Diophantine equation x 4 + 2y 4 = z 4 + 4w 4 admits essentially one and only one nontrivial solution, namely (±1 484 801, ±1 203 120, ±1 169 407, ±1 157 520). The investigation is based on a systematic search by computer.