On Fermat's marginal note: a suggestion


Or: Fermat and the Cubic Roots of Unity
Nico F. Benschop . . . n.benschop@chello.nl . . . NMC33, U-Twente, 17-apr-1998

A suggestion is put forward regarding a partial proof of FLT case_1 which is elegant and simple enough to have caused Fermat's enthusiastic remark in the margin of his Bachet edition of Diophantus' "Arithmetica". It is based on an extension of Fermat's Small Theorem (FST) to mod p^k for any k > 0, and the cubic roots of 1 mod p^k for primes p = 1 mod 6. For this solution in residues the exponent p distributes over a sum, which blocks extension to equality for integers, providing a partial proof of FLT case_1 for all p = 6m+1. This approach is consistent with Fermat's insights in arithmetic, at the time (1637) he discovered his small theorem on p-th power residues mod p.

Note: In a companion paper [1], on the triplet structure of Arithmetic mod p^k, this cubic root solution is extended to the general rootform of FLT mod p^k (case1), called "triplet". The cubic root solution involves one inverse pair: a+a^{-1} == -1 mod p^k (a^3=1 mod p^k), a triplet has three inverse pairs in a 3-loop: a+b^{-1} = b+c^{-1} = c+a^{-1} = -1 mod p^k where abc==1 mod p^k, which is not restricted to p-th power residues (for some p \geq 59) but applies to all residues in the group G_k(.) of units in the semigroup of multiplication mod p^k.



Keywords : Marginal note, Fermat Last Theorem, FST, FLT, Bachet, Diophantus, Pythagoras, cubic roots of unity, residue arithmetic, integer powersum.

NMC-33(.htm) . . . individual-talks program Neth.Math.Congress'98 U-Twente, Enschede, NL
full text (.dvi) . . . 6 pgs, 32k - (c) 17apr'98 - full text (.pdf)
intro (.htm) . . . "A day with Fermat in Toulouse" (1640)
[1] nfb0(.pdf) . . . "Triplets as additive structure of units group mod p^k ...."
published in Acta Methematica of the Univ. Bratislava (nov 2005)
as: "Additive structure of the group of units mod p^k,
with Core and Carry concepts for extension to integers"
AMUC 2005, V2 (pg 169 - 184)
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