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A018786
Numbers that are the sum of two 4th powers in more than one way.
10
635318657, 3262811042, 8657437697, 10165098512, 51460811217, 52204976672, 68899596497, 86409838577, 138519003152, 160961094577, 162641576192, 264287694402, 397074160625, 701252453457, 823372979472, 835279626752
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.
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).
Sequence in context: A343077 A374579 A360306 * A003824 A348655 A105382
KEYWORD
nonn
STATUS
approved