OFFSET
1,1
COMMENTS
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
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, D1.
LINKS
Mia Muessig, Table of n, a(n) for n = 1..30000 (terms 1..111 from Vincenzo Librandi, terms 112..4359 from Sean A. Irvine)
J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
Mia Muessig, Julia code for finding general taxicab numbers
Eric Weisstein's World of Mathematics, Biquadratic Number.
Eric Weisstein's World of Mathematics, Diophantine Equation.
FORMULA
A weak lower bound: a(n) >> n^2. - Charles R Greathouse IV, Jul 12 2024
EXAMPLE
a(1) = 59^4 + 158^4 = 133^4 + 134^4.
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
MATHEMATICA
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved