OFFSET
3,1
COMMENTS
a(6) and a(7) are only conjectures; the earlier terms have (apparently) been proved.
LINKS
A. Bremner and P. Morton, A new characterization of the integer 5906, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.
Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Alexis Newton and Jeremy Rouse, Integers that are sums of two rational sixth powers, arXiv:2101.09390 [math.NT], 2021.
Dave Rusin, Seeking counterexamples to FLT in other rings [Broken link]
Dave Rusin, Seeking counterexamples to FLT in other rings [Cached copy]
EXAMPLE
a(3) = 6 = (17/21)^3 + (37/21)^3
a(4) = 5906 = (25/17)^4 + (149/17)^4
a(5) = 68101 = (15/2)^5 + (17/2)^5
a(6) <= 164634913 = (44/5)^6 + (117/5)^6 (John W. Layman, Oct 20 2005)
a(7) <= 69071941639 = (63/2)^7 + (65/2)^7
From a posting to the Number Theory Mailing List by Seiji Tomita (fermat(AT)M15.ALPHA-NET.NE.JP), Sep 10 2009: (Start)
a(8) <= (50429/17)^8 + (43975/17)^8
a(9) <= (257/2)^9 + (255/2)^9
a(10) <= (1199/5)^10 + (718/5)^10
a(11) <= (1025/2)^11 + (1023/2)^11
a(12) <= (9298423/17)^12 + (8189146/17)^12
a(13) <= (4097/2)^13 + (4095/2)^13
a(14) <= (76443/5)^14 + (16124/5)^14
a(15) <= (16385/2)^15 + (16383/2)^15
a(16) <= (3294416782861362/97)^16 + (2731979866522411/97)^16
a(17) <= (65537/2)^17 + (65535/2)^17
a(18) <= (1721764/5)^18 + (922077/5)^18
a(19) <= (262145/2)^19 + (262143/2)^19
a(20) <= (726388197629/17)^20 + (86503985645/17)^20
(End)
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
David W. Wilson, Oct 19 2005
STATUS
approved