

A088703


Numbers of form x^5 + y^5, x,y > 0 and x<>y.


7



33, 244, 275, 1025, 1056, 1267, 3126, 3157, 3368, 4149, 7777, 7808, 8019, 8800, 10901, 16808, 16839, 17050, 17831, 19932, 24583, 32769, 32800, 33011, 33792, 35893, 40544, 49575, 59050, 59081, 59292, 60073, 62174, 66825, 75856, 91817
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OFFSET

1,1


COMMENTS

Up to n = 100000, no instances occur where n is the sum of two distinct 5th powers in two different ways. Conjecture: no number can be expressed as the sum of two 5th powers in more than one way.
The lower bound on the solution to N = a^5 + b^5 = c^5 + d^5 is N > 1.02 * 10^26. [Balarka Sen, Oct 24 2013]
The lower bound on the solution with 2 distinct representations is 4.01*10^30 [Ekl, Table 9].  R. J. Mathar, Sep 07 2017


REFERENCES

Guy, Richard K., Unsolved Problems in Number Theory, 2nd Ed., SpringerVerlag(1994), pp. 140.


LINKS

Table of n, a(n) for n=1..36.
R. L. Ekl, New results in equal sums of like powers, Math. Comp. 67 (1998) 13091315.
Wikipedia, Generalized Taxicab Numbers


EXAMPLE

33 = 2^5 + 1^5, so 33 is in sequence. 64 = 2^5 + 2^5 is not.


MATHEMATICA

lst={}; e=5; Do[Do[x=a^e; Do[y=b^e; If[x+y==n, AppendTo[lst, n]], {b, Floor[(nx)^(1/e)], a+1, 1}], {a, Floor[n^(1/e)], 1, 1}], {n, 8!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 22 2009 *)
Union[#[[1]]^5+#[[2]]^5&/@Subsets[Range[10], {2}]] (* Harvey P. Dale, Apr 25 2012 *)


PROG

(PARI) powers2(m1, m2, p1) = { for(k=m1, m2, a=powers(k, p1); if(a==1, print1(k", ")) ); } powers(n, p) = { z1=0; z2=0; c=0; cr = floor(n^(1/p)+1); for(x=1, cr, for(y=x+1, cr, z1=x^p+y^p; if(z1 == n, c++); ); ); return(c) }


CROSSREFS

Subsequence of A003347.
Sequence in context: A274639 A306879 A178448 * A321561 A034679 A017673
Adjacent sequences: A088700 A088701 A088702 * A088704 A088705 A088706


KEYWORD

nonn


AUTHOR

Cino Hilliard, Nov 22 2003


EXTENSIONS

Edited by Ralf Stephan, Dec 30 2004


STATUS

approved



