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Search: id:A088719
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| A088719 |
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Numbers that can be represented as a^7+b^7, with 0<a<b, in exactly one way. |
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+0
6
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| 129, 2188, 2315, 16385, 16512, 18571, 78126, 78253, 80312, 94509, 279937, 280064, 282123, 296320, 358061, 823544, 823671, 825730, 839927, 901668, 1103479, 2097153, 2097280, 2099339, 2113536, 2175277, 2377088, 2920695
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Conjecture: no number can be expressed as such a sum in more than one way.
No solutions to the 7.2.2 (A^7 + B^7 = C^7 + D^7), 7.2.3, 7.2.4, or 7.2.5 equations are known. The smallest 7.2.6 equation is: 125^7 + 24^7 = 121^7 + 94^7 + 83^7 + 61^7 + 57^7 + 27^7 = 476841744674549. - Jonathan Vos Post (jvospost3(AT)gmail.com), May 04 2006
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REFERENCES
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Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309-1315, 1998.
Ekl, R. L. "Equal Sums of Four Seventh Powers." Math. Comput. 65, 1755-1756, 1996.
Sastry, S. and Rai, T. "On Equal Sums of Like Powers." Math. Student 16, 18-19, 1948.
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LINKS
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Eric Weisstein's World of Mathematics, Diophantine Equation: 7th Powers
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EXAMPLE
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129 = 1^7+2^7.
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MATHEMATICA
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lst={}; e=7; Do[Do[x=a^e; Do[y=b^e; If[x+y==n, AppendTo[lst, n]], {b, Floor[(n-x)^(1/e)], a+1, -1}], {a, Floor[n^(1/e)], 1, -1}], {n, 3*8!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 23 2009]
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PROGRAM
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(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) }
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CROSSREFS
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Cf. A003369.
Sequence in context: A043383 A036548 A046286 this_sequence A034681 A017677 A013955
Adjacent sequences: A088716 A088717 A088718 this_sequence A088720 A088721 A088722
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KEYWORD
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nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Nov 22 2003
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EXTENSIONS
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Edited by Don Reble (djr(AT)nk.ca), May 03 2006
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