

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
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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), 778780.
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]] (* JeanFrançois 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(tc^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



