

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.


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
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OFFSET

1,1


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 order8 element of the multiplicative group (Z/pZ)*, which has p1 elements. Therefore p is congruent to 1 mod 8.


LINKS

Table of n, a(n) for n=1..30.
A. Bremner and P. Morton, A new characterization of the integer 5906, Manuscripta Math. 44 (1983) 187229; Math. Rev. 84i:10016.
Steven R. Finch, On a generalized FermatWiles equation [broken link]
Steven R. Finch, On a Generalized FermatWiles Equation [From the Wayback Machine]
Eric Weisstein's World of Mathematics, Biquadratic Number


FORMULA

Numbers in A060387 but not in A003336.


EXAMPLE

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


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]]]


CROSSREFS

Cf. A060387, A003336, A020897.
Sequence in context: A251469 A025513 A015295 * A163209 A216942 A252289
Adjacent sequences: A209428 A209429 A209430 * A209432 A209433 A209434


KEYWORD

nonn


AUTHOR

JeanFrançois Alcover, Mar 09 2012


EXTENSIONS

Definition corrected by Hugo Pfoertner, Nov 08 2016


STATUS

approved



