40 ARITHMETICAL RECREATIONS [CH. II 



he asserted that all numbers of the form 2 m + 1, where m = 2 n , 

 are primes*, but he added that, though he was convinced of 

 the truth of this proposition, he could not obtain a valid 

 demonstration. 



It may be shown that 2 m + 1 is composite if m is not a power 

 of 2, but of course it does not follow that 2 m + 1 is a prime 

 if m is a power of 2, say, 2". As a matter of fact the theorem 

 is not true. In 1732 Eulerf showed that if n = 5 the formula 

 gives 4294,967297, which is equal to 641 x 6,700417 : curiously 

 enough, these factors can be deduced at once from Fermat's 

 remark on the possible factors of numbers of the form 2 m ± 1, 

 from which it may be shown that the prime factors (if any) 

 of 2 s1 + 1 must be primes of the form 64rj, + 1. 



During the last thirty years it has been shown* that the 

 resulting numbers are composite when n = 6, 7, 8, 9, 11, 12, 18, 

 23, 36, 38 and 73. The digits in the last of these numbers are 

 so numerous that, if the number were printed in full with the 

 type and number of pages used in this book, many more 

 volumes would be required than are contained in all the public 

 libraries in the world. I believe that Eisenstein asserted that 

 the number of primes of the form 2 m + 1, where m = 2™, is 

 infinite: the proof has not been published, but perhaps it 

 might throw some light on the general theory. 



Fermat's Last Theorem. I pass now to another assertion 

 made by Fermat which hitherto has not been proved. This, 

 which is sometimes known as Fermat's Last Theorem, is to the 

 effect § that no integral values of x, y, z can be found to satisfy 



* Letter of Oct. 18, 1640, Opera, Toulouse, 1679, p. 162: or Braaainne's 

 Precis, p. 143. 



t Commentarii Academiae Scientiarum Petropolitanae, Petrograd, 1738, 

 vol. vi, p. 104; aee also Novi Comm. Acad. Sci. Petrop., Petrograd, 1764, 

 vol. rx, p. 101 : or Commentationes Arithvieticae Gollectae, Petrograd, 1849, 

 vol. i, pp. 2, 357. 



% For the factors and bibliographical references, see A. J. C. Cunningham 

 and A. E. Western, Transactions of the London Mathematical Society, 1903, 

 series 2, vol. i, p. 175 ; and J. C. Morehead and A. E. Western, Bulletin of the 

 American Mathematical Society, 1909, vol. xvi, pp. 1 — 6. 



§ Fermat's enunciation will be found in his edition of Diophantus, Toulouse, 

 1670, bk. ii, qu. 8, p. 61; or Brassinne's Precis, Paris, 1853, p. 53. For 



