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%I #41 Apr 14 2019 14:21:44
%S 5906,469297,926977,952577,1127857,1298257,1347361,1647377,2455361,
%T 3342817,4928977,5268706,5519537,8588161,8879537,9339361,9391537,
%U 9846017,11414017,14543026,15547297,16502722,16657217,16672322,16830017,19730162,23672002,25030097,27681937,27979762
%N 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.
%C 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.
%C 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.
%H A. Bremner and P. Morton, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002225409">A new characterization of the integer 5906</a>, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/fermat.pdf">On a generalized Fermat-Wiles equation</a> [broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010602030546/http://www.mathsoft.com/asolve/fermat/fermat.html">On a Generalized Fermat-Wiles Equation</a> [From the Wayback Machine]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BiquadraticNumber.html">Biquadratic Number</a>
%F Numbers in A060387 but not in A003336.
%e 5906 is in the sequence because a^4 + b^4 = 5906*c^4 has the solution (a,b,c) = (25,149,17).
%t 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]]]
%Y Cf. A060387, A003336, A020897.
%K nonn
%O 1,1
%A _Jean-François Alcover_, Mar 09 2012
%E Definition corrected by _Hugo Pfoertner_, Nov 08 2016
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