A060387 Numbers k such that x^4 + y^4 = k * z^4 is solvable in nonzero integers x,y,z.

2, 17, 32, 82, 97, 162, 257, 272, 337, 512, 626, 641, 706, 881, 1250, 1297, 1312, 1377, 1552, 1921, 2402, 2417, 2482, 2592, 2657, 3026, 3697, 4097, 4112, 4177, 4352, 4721, 4802, 5392, 5906

Offset: 1

Author

Michel ten Voorde, Apr 04 2001

Keywords

• nonn
• nice
• more

Comments

Numbers k such that k = x^4 + y^4 has a solution in positive rationals x, y.

Links

• A. Bremner and P. Morton, A new characterization of the integer 5906, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.
• Steven R. Finch, On a generalized Fermat-Wiles equation [broken link]
• Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
• Eric Weisstein's World of Mathematics, Biquadratic Number

Crossrefs

5906 is the first term not in A003336. Cf. A020897.

Cf. A111152, A209431.

Programs

Mathematica

r[n_, z_] := Reduce[0 < x <= y && x^4 + y^4 == n*z^4, {x, y}, Integers]; zm[] = 1; zm[5906] = 17; ok[n] := (tf = False; Do[ If[ r[n, z] =!= False, tf = True; Break[]], {z, 1, zm[n]}]; tf); A060387 = Reap[ Do[ If[ ok[n], Print[n]; Sow[n]], {n, 1, 5906}]][[2, 1]](* Jean-François Alcover, Mar 09 2012 *)