A209431 Numbers n such that x^4 + y^4 = n * z^4 is solvable in nonzero integers x,y,z with z > 1 and gcd(x,y,n) = 1.

5906, 469297, 926977, 952577, 1127857, 1298257, 1347361, 1647377, 2455361, 3342817, 4928977, 5268706, 5519537, 8588161, 8879537, 9339361, 9391537, 9846017, 11414017, 14543026, 15547297, 16502722, 16657217, 16672322, 16830017, 19730162, 23672002, 25030097, 27681937, 27979762

Offset: 1

Author

Jean-François Alcover, Mar 09 2012

Keywords

• nonn

Comments

Values of z (1, 17, 41, 73, 89, ...) are elements of sequence A004625 (divisible only by primes congruent to 1 mod 8). The first composite z is 697 = 17*41: 41^4 + 822091^4 = 1935300738962*697^4.

Proof (after Ms. Adina Calvo) that values of z are divisible only by primes congruent to 1 mod 8: Let {x,y,z} be a nontrivial solution and p an odd prime divisor of z. Reducing the equation mod p, one gets in Z/pZ: x^4 + y^4 = 0 mod p. Hence (x*y^-1)^4 = -1, then x*y^-1 is an order-8 element of the multiplicative group (Z/pZ)*, which has p-1 elements. Therefore p is congruent to 1 mod 8.

Examples

			5906 is in the sequence because a^4 + b^4 = 5906*c^4 has the solution (a,b,c) = (25,149,17).
		

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

Cf. A060387, A003336, A020897.

Programs

Mathematica

BiquadraticFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 4]} & /@ FactorInteger[n]); max = 10000; Sort[ Reap[Do[nz4 = x^4 + y^4; z4 = nz4/BiquadraticFreePart[nz4]; z = z4^(1/4); n = nz4/z4; If[z4 > 1 && IntegerQ[z] && GCD[x, y, z] == 1, Print[{n, x, y, z}]; Sow[n]], {x, 1, max}, {y, x, max}]][[2, 1]]]

Formula

Numbers in A060387 but not in A003336.

Extensions

Definition corrected by Hugo Pfoertner, Nov 08 2016