ON THE THEORY OF NUMBERS. 149 



has left us a proof of the impossibility of this equation in the case of n=i; 

 by a method which Euler has extended to the case of ?*=3, we may suppose, 

 without loss of generality, that 71 is an uneven prime A greater than 3, and we 



plicable, if his conclusions reposed on induction only, that he should never have adopted an 

 erroneous generalization ; and yet, with the exception of the " Last Theorem " (the demon- 

 stration of which, after two centuries, is still incomplete), every proposition of Fermat's has 

 been verified by the labours of his successors. There is, indeed, one other exception to this 

 statement; but it is an exception which proves the rule. In the letter to Sir Kenelm Digby 

 which concludes the ' Commercium Epistolieum, etc' edited by Wallis (Oxford, 1658), 



Fermat enuntiates the proposition that the numbers contained in the formula 2 2 +1 are all 

 primes, acknowledging, however, that, though convinced of its truth, he had not succeeded 

 in obtaining its demonstration. This letter, which is undated, was written in 1C58 ; but it 

 appears, from a letter of Fermat's to M. de * * *, dated October 18, 1640, that even at that 

 earlier date he was acquainted with the proposition, and had convinced himself of its trutli 

 (D. Petri de Fermat Varia Opera Mathematica, Tolosse, 1679, p. 162). It was, however, 



subsequently observed by Euler that 22 5 +l=4294967297 = 641x6700417, i. e. that the 

 undemonstrated proposition is untrue (Op. Arith. collecta, vol. i. p. 356). The error, if it is 

 an error, is a fortunate one for Fermat; it exemplifies his candour and veracity, and it shows 

 that he did not mistake inductive probability for rigorous demonstration : — " Mais je vous 

 advoue tout net," are his words in the letter last referred to, " (car par advance je vous ad- 

 vertis que comme je ne suis pas capable de m'attribuer plus que je ne scay, je dis avec merne 

 franchise ce que je ne scay pas) que je n'ay peu encore demonstrer l'exclusion de tous divi- 

 seurs en cette belle proposition que je vous avois envoyee, et que vous m'avez confermee 

 touchant les nombres 3, 5, 17, 257, 6553, &c. Car bien que je reduise l'exclusion a la 

 pluspart des nombres, et que j'aye meme des raisons probables pour le reste, je n'ay peu 

 encore demonstrer necessairement la verite de cette proposition, de laquelle pourtant je ne 

 doute non plus a cette heure que je faisois auparavant. Si vous en avez la preuve assuree, 

 vous m'obligerez de me la communiquer : car aprcs cela rien ne m'arrestera en ces matieres." 



The " Last Theorem " is enunciated by Fermat as follows : — 



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

 generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est 

 dividere; cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non 

 caperet." (Fermat's Diophantus, p. 51.) 



Fermat has also asserted that neither the sum (ibid. p. 258) nor the difference (ibid. p. 338) 

 of two biquadrates can be a square. Each of these propositions comprehends the theorem 

 that the sum of two biquadrates cannot be a biquadrate ; and of the second, we possess 

 a very remarkable demonstration by Fermat himself (ibid. p. 338 ; and compare Euler, 

 Elemens d'Algubre, vol. ii. sect. 13; Legendre, Theorie des Nombres, vol. ii. p. 1). The 

 essential part of this demonstration consists in showing that, from any supposed solution 

 of the Diophantine equation <r 4 — y 4 = a square, another solution maybe deduced in which 

 the values of the indeterminates are not equal to zero, and yet are absolutely less than in 

 the proposed solution, from which it immediately follows that the Diophantine equation 

 is impossible. This method has been successfully employed by Eider (joe. cit.) to demon- 

 strate several negative Diophantine propositions, and in particular the theorem that the sum 

 of two cubes cannot be a cube. The only arithmetical principles (not included in the first 

 elements of the science) which are employed by Euler and Fermat in their applications of 

 this method, relate to certain simple properties of the quadratic forms x"-\-y", x 2 -\-2y 2 , 

 • r2 +3y 2 ; and as these principles seem inadequate to overcome the difficulties presented by 

 the equation x n -\-y"-\-z n =0, when n is > 4, it is probable that Fermat's " demonstratio 

 mirabilis sane " of the general theorem was entirely different from that which he has inci- 

 dentally given of the particular case. 



The impossibility of the equation ,r n -(-y"+z n = for n = 5 was first demonstrated by Le- 

 gendre (Memoires de l'Academie des Sciences, 1823, vol. vi. p. 1, or Theorie des Nombres, 

 vol. ii. p. 361. See also an earlier paper by Lejeune Diricblet, Crelle, vol. iii. p. 354, with 

 the addition at p. 368, and a later one by M. Lebesgue, Liouville, vol. viii. p. 49) ; for n = 14, 

 by Diricblet (Crelle, vol. ix. p. 390); and for re=7, by M. Lame (Memoires des Savans 

 Etrangers, vol. viii. p. 421, or Liouville, vol. v. p. 195. See also the Comptes Kendus, vol. ix. 

 p. 359, and a paper by M. Lebesgue, Liouville, vol. v. pp. 276 & 348). But the methods 

 employed in these researches are specially adapted to the particular exponents considered, 

 and do not seem likely to supply a general demonstration. The proof in Barlow's Theory of 

 Numbers, pp. 160-169, is erroneous, as it reposes (see p. 168) on an elementary proposition 

 (cor. 2, p. 20) which is untrue. A memoir by M. Kummer on the equation x' M -j-y 2A =z 2A -, 

 in which complex numbers are not employed, and in which no single case of the theorem is 



