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A018786 Numbers that are the sum of two 4th powers in more than one way. 5
635318657, 3262811042, 8657437697, 10165098512, 51460811217, 52204976672, 68899596497, 86409838577, 138519003152, 160961094577, 162641576192, 264287694402, 397074160625, 701252453457, 823372979472, 835279626752 (list; graph; refs; listen; history; text; internal format)
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

Vincenzo Librandi, Table of n, a(n) for n = 1..111

J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.

Eric Weisstein's World of Mathematics, Biquadratic Number.

Eric Weisstein's World of Mathematics, Diophantine Equation.

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

CROSSREFS

Cf. A003824.

Sequence in context: A237364 A251498 A233848 * A003824 A105382 A032432

Adjacent sequences:  A018783 A018784 A018785 * A018787 A018788 A018789

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified May 25 14:13 EDT 2017. Contains 287038 sequences.