On the sums of arbitrary different biquadrates in two different ways

EasyChair Preprint no. 2433

Abstract

‎The quartic Diophantine equation $A^4+hB^4=C^4+hD^4$‎, ‎where $h$ is a fixed arbitrary positive integer‎, ‎has been studied by some mathematicians‎.

‎In a recent paper‎, ‎we studied this equation by using elliptic curve theory‎, ‎and worked out some solutions of this equation for certain values of $h$‎, ‎in particular for the values which has not already been found a solution in the range where $A‎, ‎B‎, ‎C‎, ‎D \le 100000$ by computer search‎. ‎Finally we presented two conjecture such that if one of them is correct then we may solve this equation for every rational number $h$‎.

‎In the present paper‎, ‎we use the solutions of aforementioned Diophantine equation‎, ‎as well as a simple idea to show that how some numbers can be written as the sums of two‎, ‎three‎, ‎four‎, ‎five‎, ‎or more different biquadrates in two different ways‎. ‎In particular we give examples for the sums of $2$‎, ‎$3$‎, ‎$\cdots$‎, ‎and $10$‎, ‎biquadrates expressed in two different ways‎.

Keyphrases: Diophantine equations‎, ‎Biquadrates‎, ‎Elliptic curve, ‎Quartic Diophantine equations‎

@Booklet{EasyChair:2433,
  author = {Mehdi Baghalaghdam and Farzali Izadi},
  title = {On the sums of arbitrary different biquadrates  in two different ways},
  howpublished = {EasyChair Preprint no. 2433},

  year = {EasyChair, 2020}}