%I #63 Jul 12 2024 14:23:35

%S 635318657,3262811042,8657437697,10165098512,51460811217,52204976672,

%T 68899596497,86409838577,138519003152,160961094577,162641576192,

%U 264287694402,397074160625,701252453457,823372979472,835279626752

%N Numbers that are the sum of two 4th powers in more than one way.

%C Since 4th powers are squares, this is a subsequence of A024508, the analog for squares. Sequence A001235 is the analog for third powers (taxicab numbers). Sequence A255351 lists max {a,b,c,d} where a^4 + b^4 = c^4 + d^4 = a(n), while A255352 lists the whole quadruples (a,b,c,d). - _M. F. Hasler_, Feb 21 2015

%D R. K. Guy, Unsolved Problems in Number Theory, D1.

%H Mia Muessig, <a href="/A018786/b018786.txt">Table of n, a(n) for n = 1..30000</a> (terms 1..111 from Vincenzo Librandi, terms 112..4359 from Sean A. Irvine)

%H J. Leech, <a href="http://dx.doi.org/10.1017/S0305004100032850">Some solutions of Diophantine equations</a>, Proc. Camb. Phil. Soc., 53 (1957), 778-780.

%H Mia Muessig, <a href="https://github.com/PhoenixSmaug/taxicab-numbers">Julia code for finding general taxicab numbers</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BiquadraticNumber.html">Biquadratic Number.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DiophantineEquation4thPowers.html">Diophantine Equation.</a>

%F A weak lower bound: a(n) >> n^2. - _Charles R Greathouse IV_, Jul 12 2024

%e a(1) = 59^4 + 158^4 = 133^4 + 134^4.

%e a(2) = 7^4 + 239^4 = 157^4 + 227^4. Note the remarkable coincidence that here all of {7, 239, 157, 227} are primes. The next larger solution with this property is 17472238301875630082 = 62047^4 + 40351^4 = 59693^4 + 46747^4. - _M. F. Hasler_, Feb 21 2015

%t Select[ Split[ Sort[ Flatten[ Table[x^4 + y^4, {x, 1, 1000}, {y, 1, x}]]]], Length[#] > 1 & ][[All, 1]] (* _Jean-François Alcover_, Jul 26 2011 *)

%o (PARI) n=4;L=[];for(b=1,999,for(a=1,b,t=a^n+b^n;for(c=a+1,sqrtn(t\2,n),ispower(t-c^n,n)||next;print1(t",")))) \\ _M. F. Hasler_, Feb 21 2015

%o (PARI) list(lim)=my(v=List()); for(a=134,sqrtnint(lim,4)-1, my(a4=a^4); for(b=sqrtnint((4*a^2 + 6*a + 4)*a,4)+1,min(sqrtnint(lim-a4,4),a), my(t=a4+b^4); for(c=a+1,sqrtnint(lim,4), if(ispower(t-c^4,4), listput(v,t); break)))); Set(v) \\ _Charles R Greathouse IV_, Jul 12 2024

%Y Subsequence of A003336 (and hence A004831) and A024508 (and hence A001481).

%Y Cf. A001235, A003336, A003824, A309762.

%K nonn

%O 1,1

%A _David W. Wilson_