CH. Il] ARITHMETICAL RECREATIONS 41 



the equation x n + y n = z n , if n is an integer greater than 2. 

 This proposition has acquired extraordinary celebrity from the 

 fact that no general demonstration of it has been given, but 

 there is no reason to doubt that it is true. 



Fermat seems to have discovered its truth first * for the case 

 n = 3, and then for the case n = 4. His proof for the former 

 of these cases is lost, but that for the latter is extantf , and a 

 similar proof for the case of n = 3 was outlined by Euler J . These 

 proofs depend upon showing that, if three integral values of 

 x, y, z can be found which satisfy the equation, then it will be 

 possible to find three other and smaller integers which also 

 satisfy it : in this way finally we show that the equation must 

 be satisfied by three values which obviously do not satisfy it. 

 Thus no integral solution is possible. This method is inapplicable 

 to the general case. 



Fermat 's discovery of the general theorem was made later. 

 A proof can be given on the assumption that every number can 

 be resolved into the product of powers of primes in only one way. 

 The assumption is true of real integers, but is not necessarily 

 true for algebraic integers — an algebraic integer being defined 

 as a root of an equation 



x n + a 1 x T ^- 1 +... + a n = 0, 

 whose coefficients, a, are arithmetical integers; for instance, 

 a .J. l V _ m , where a, b, and m, are arithmetical integers, is an 

 algebraic integer. Thus, admitting the use of these generalized 

 integers, 21 can be expressed in three ways as the product 

 of primes, namely, 3 and 7, or of 4 + V - 5 and 4 - V - 5, or 

 of 1 + 2 V^5 and 1-2 V^5 ; and similarly, there are values 

 of n for which Fermat's equation leads to expressions which 

 can be factorized in more than one way. It is possible that 

 Fermat's argument rested on the above erroneous supposition, 

 but this is an unsupported conjecture. At any rate he asserted 

 bibliographical references, see L. E. Dickson, History of the Theory of Numbers, 

 Washington, 1920, vol. n, ch. 26 ; see also L. J. Mordell, Fermat's Last Theorem, 

 Cambridge, 1921. 



* See a Letter from Fermat quoted in my History of Mathematics, London, 



ohapter xv. 



t Fermat's Diophantus, note on p. 339; or Brassinne's Precis, p. 127. 



J Euler's Algebra (English trans. 1797), vol. ii, ehap. sv, p. 247 : one point 

 was overlooked by Euler, but the omission can be supplied. 



