Search: id:A060748
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%I A060748
%S A060748 1,6,19,657,21691,489489,9902523,1144421889,1683200989470,
%T A060748 349043376293530
%N A060748 Smallest m such that x^3+y^3=m has rank n.
%C A060748 Nick Rogers (rogers(AT)fas.harvard.edu), Jul 03 2003: I have verified 
               that the first 5 entries are correct; the first two are basically 
               trivial and the third is due to Selmer. I'm not sure who first discovered 
               entries 4 and 5 and I expect that they had been previously proved 
               to be the smallest values, (cont.)
%C A060748 but I have rechecked that they are minimal for their respective rank 
               using a combination of 3-descent, MAGMA and John Cremona's program 
               mwrank. (cont.)
%C A060748 There are new smaller values for ranks 6 and 7, namely k = 9902523 has 
               rank 6 and k = 1144421889 has rank 7. 3-descent combined with Ian 
               Connell's package apecs for Maple verifies that these are minimal 
               subject to the Birch and Swinnerton-Dyer conjecture and the Generalized 
               Riemann Hypothesis for L-functions associated to elliptic curves. 
               (cont.)
%C A060748 Finally, there are new entries for ranks 8 and 9: k = 1683200989470 has 
               rank 8 and k = 148975046052222390 has rank 9. It seems somewhat likely 
               that the rank 8 example is minimal. (end.)
%D A060748 Noam D. Elkies, Posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), 
               Oct 19 2003, for a(9)
%D A060748 Noam D. Elkies and Nicholas F. Rogers, Posting to Number Theory List 
               (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Jul 18 2003, for a(8) and a(9).
%D A060748 Troy Kessler (kesslert(AT)surfree.com), Posting to Number Theory List 
               (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Apr 22, 2001.
%D A060748 Nick Rogers, Rank computations for the congruent number elliptic curves. 
               Experimental Mathematics 9 (2000), no. 4, 591-594
%H A060748 Experimental Mathematics, <a href="http://www.expmath.org/">Home Page</
               a>
%Y A060748 Cf. A060838.
%Y A060748 Sequence in context: A118411 A091876 A041066 this_sequence A075251 A090590 
               A002566
%Y A060748 Adjacent sequences: A060745 A060746 A060747 this_sequence A060749 A060750 
               A060751
%K A060748 nonn,nice
%O A060748 0,2
%A A060748 N. J. A. Sloane (njas(AT)research.att.com), Apr 23 2001

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