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XXVII. On Fermafs Theorem. By H. F. Talbot, Esq., F.R.S., &c. 



(Read 7th April 1856.) 



It is well known that no satisfactory demonstration has ever been given of 

 Fermat's celebrated theorem, which asserts that the equation a n =b n + c n is impos- 

 sible, if a, b, c, are whole numbers, and n is any whole number greater than 2. 

 In Legendre's TMorie des Nombres, he demonstrates the cases of ft =3, n=4, and 

 « =5, the latter only in his Second Supplement. In Crelle's MathematicalJour- 

 nal, ix. 390, M. Dirichlet, a mathematician of Berlin, has demonstrated the case 

 of ft =14, but I am not aware whether his demonstration is considered successful. 

 Legendre informs us (Second Supplement, p. 3) that the Academy of Sciences, 

 with the view of doing honour to the memory of Fermat, proposed, as the sub- 

 ject of one of its mathematical prizes, the demonstration of this theorem ; but 

 the Concourse, though prolonged beyond the usual term, produced no result. 



It is a remarkable circumstance, however, that Fermat himself was in pos- 

 session of the demonstration, or at least believed himself to be so, and he describes 

 his demonstration as being a wonderful one — mirabilem sane.* He does not say that 

 the theorem itself is wonderful, but his demonstration of it ; from which I think 

 it likely that he meant to say that it was very remarkable for its shortness and 

 simplicity. 



Since, however, subsequent mathematicians have failed to discover any de- 

 monstration, much less an extremely simple one, of this celebrated theorem, it 

 has been surmised that Fermat deceived himself in this matter, and that his 

 demonstration, if it had been preserved to us, would have proved unsatisfactory. 

 Legendre says, "Fermat a pu se meprendre sur l'exactitude ou la generalite de 

 sa demonstration." 



Nevertheless, in considering this question attentively, I have found that there 

 is one case in which Fermat's theorem admits of a singularly simple demonstra- 

 tion ; and as I do not find it noticed in any mathematical work to which I have 

 been able to refer, I think it worthy of being brought under the notice of mathe- 

 maticians. It may possibly prove to be a step in the right direction towards the 

 recovery of Fermat's lost demonstration. It is, moreover, in itself a very ex- 

 tended and remarkable theorem, although less so than that of Fermat. 



* " Cubum autem in duos cubos aut quadrato-quadratum in duos quadrato-quadratos et gene- 

 raliter nullam in infinitum, ultra quadrature, potestatem in duos ejusdem nominis fas est dividere. 

 Cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet." — Fermat, 

 Notes sur Diophante, p. 61. 



VOL. XXI. PART III. 5 R 



