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%I #26 Feb 25 2023 06:07:24
%S 6,5906,68101,164634913,69071941639
%N Smallest number that is a sum of two n-th powers of positive rationals but not of two n-th powers of positive integers.
%C a(6) and a(7) are only conjectures; the earlier terms have (apparently) been proved.
%H A. Bremner and P. Morton, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002225409">A new characterization of the integer 5906</a>, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/fermat.pdf">On a Generalized Fermat-Wiles Equation</a> [broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010602030546/http://www.mathsoft.com/asolve/fermat/fermat.html">On a Generalized Fermat-Wiles Equation</a> [From the Wayback Machine]
%H Alexis Newton and Jeremy Rouse, <a href="https://arxiv.org/abs/2101.09390">Integers that are sums of two rational sixth powers</a>, arXiv:2101.09390 [math.NT], 2021.
%H Dave Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/00_incoming/flt_rings">Seeking counterexamples to FLT in other rings</a> [Broken link]
%H Dave Rusin, <a href="/A111152/a111152.txt">Seeking counterexamples to FLT in other rings</a> [Cached copy]
%e a(3) = 6 = (17/21)^3 + (37/21)^3
%e a(4) = 5906 = (25/17)^4 + (149/17)^4
%e a(5) = 68101 = (15/2)^5 + (17/2)^5
%e a(6) <= 164634913 = (44/5)^6 + (117/5)^6 (_John W. Layman_, Oct 20 2005)
%e a(7) <= 69071941639 = (63/2)^7 + (65/2)^7
%e From a posting to the Number Theory Mailing List by Seiji Tomita (fermat(AT)M15.ALPHA-NET.NE.JP), Sep 10 2009: (Start)
%e a(8) <= (50429/17)^8 + (43975/17)^8
%e a(9) <= (257/2)^9 + (255/2)^9
%e a(10) <= (1199/5)^10 + (718/5)^10
%e a(11) <= (1025/2)^11 + (1023/2)^11
%e a(12) <= (9298423/17)^12 + (8189146/17)^12
%e a(13) <= (4097/2)^13 + (4095/2)^13
%e a(14) <= (76443/5)^14 + (16124/5)^14
%e a(15) <= (16385/2)^15 + (16383/2)^15
%e a(16) <= (3294416782861362/97)^16 + (2731979866522411/97)^16
%e a(17) <= (65537/2)^17 + (65535/2)^17
%e a(18) <= (1721764/5)^18 + (922077/5)^18
%e a(19) <= (262145/2)^19 + (262143/2)^19
%e a(20) <= (726388197629/17)^20 + (86503985645/17)^20
%e (End)
%K nonn,more,hard
%O 3,1
%A _David W. Wilson_, Oct 19 2005
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