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A111152 Smallest number that is a sum of two n-th powers of positive rationals but not of two n-th powers of positive integers. 2

%I

%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 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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Last modified June 6 18:59 EDT 2020. Contains 334832 sequences. (Running on oeis4.)