A018786 Numbers that are the sum of two 4th powers in more than one way.

635318657, 3262811042, 8657437697, 10165098512, 51460811217, 52204976672, 68899596497, 86409838577, 138519003152, 160961094577, 162641576192, 264287694402, 397074160625, 701252453457, 823372979472, 835279626752, 1102393543952, 1382557417232, 1525400095457

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

The b-file of primitive solutions A003824 can be used to generate more terms than are available here. {A018786(n)} = {b*c^4 : b in A003824, c>=1}. - Martin Fuller, Feb 02 2026

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 311.

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

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

Primitive solutions A003824.

Cf. A001235, A309762.

Sequence in context: A343077 A374579 A360306 * A003824 A348655 A105382

Adjacent sequences: A018783 A018784 A018785 * A018787 A018788 A018789

Keyword

nonn

Author

David W. Wilson

Status

approved