More about four biquadrates equal one biquadrate

Abstract. A computer-generated table of the first 82 numerical solutions of A* + B* +Cl + Di = E* is presented. Some regularities are noted. Solutions of the Diophantine equation A4 + B4 + C4 + D4 = E4 in least integershave been obtained by several authors [l]-[6]. The following table includes the first82 consecutive primitive solutions of A* + B* + C4 + D4 = E4. A primitive solutionis one for which the GCD of A, B, C, D and E equals 1. The method used to findthese numerical solutions combines those of Leech [5], Ward [3] and Brudno [6].In addition, the authors used the following new restrictions:(1) The sum of two fourth powers cannot be 7, 8, or 11 (Mod 13);(2) The sum of two fourth powers cannot be 6, 7, 10, or 11 (Mod 17);(3) The sum of two fourth powers cannot be 4, 5, 6, 9, 13, 22 or 28 (Mod 29);(4) The sum of three fourth powers cannot be zero modulo 29 unless each fourthpower is zero modulo 29. The same applies to modulo 16 and modulo 5.Some of these restrictions were used by Lander [4] in his search for solutions