New Solutions to Euler's Quartic Conjecture

In 1986, Elkies disproved Euler's conjecture for fourth powers by demonstrating that the equation A 4 + B 4 + C 4 = D 4 has infinitely many integer solutions, using a method based on hyperelliptic curves. The first explicit solution is 2682440 4 + 15365639 4 + 18796760 4 = 20615673 4 , but subsequent solutions grow extremely large. After learning of Elkies' result, Frye performed an exhaustive computer search and found the smallest solution: 95800 4 + 217519 4 + 414560 4 = 422481 4 . When the search is restricted to primitive solutions whose values do not exceed 30 digits, twenty seven such solutions are presently known. In this paper, we build upon Elkies' method to obtain thirteen new primitive solutions, with values reaching up to 30 digits. Our analysis focuses on two hyperelliptic curves. The first curve gives thirteen solutions: six previously known and seven new. This group of solutions is derived from the three base solutions of Frye, MacLeod, and Tomita. The second curve gives fourteen solutions: eight known and six new. These are all generated by the three base solutions of Elkies, MacLeod, and Tomita.