740 HlSTOEY OF THE THEOKY OF NUMBERS. [CHAP. XXVI 



Lame* stated (p. 888) that Fermat's equation is impossible for a series of 

 exponents including n = 5, 11, 13. 



Lame" 47 presented his arguments in two long memoirs. 



O. Terquem 48 suggested a subscription to Lame for his 45 proof (!) 

 declaring it the greatest discovery of the century in the mathematical world. 



E. E. Kummer 49 pointed out the falsity of Lame's 45 assumption that 

 every complex integer can be decomposed into primes in a single way. 



L. WantzeP^jproved that Euclid's g.c.d. process holds for complex 

 integers a+6V 1 [already proved by C. F. Gauss 51 ] and for complex 

 integers formed from an imaginary cube root of unity, and stated that a 

 like result holds for complex integers (7), with n arbitrary, since the norm 

 (or modulus) of (7) is < 1 when a , , a n -i are between and 1 [erroneous, 

 Cauchy 62 ]. 



A. Cauchy 52 showed that the final statement by Wantzel 50 is false for 

 n = 7 and for any prime w = 4ra+1^17. He pointed out lacunae in the 

 proposed proof by Lame* 45 of Fermat's last theorem. He defined the 

 factorial of (7) to be its product by the complex members obtained from it 

 by replacing r by the remaining primitive nth roots of unity, and obtained 

 upper limits for such factorials [norms]. He 53 proved that any common 

 factor of M h = Ar h +B and M k divides M if A and B are relatively prime. 



Cauchy 54 attempted to prove the false theorem that the norm of the 

 remainder obtained on dividing one complex number (7) by another can 

 always be made less than the norm of the divisor. He concluded (falsely) 

 that a product of complex integers (7) can be decomposed into complex 

 primes in a single manner, and that the other laws of divisibility of integers 

 hold for these complex integers. 



Cauchy 55 noted (erroneous) conclusions which follow from the assump- 

 tion that his preceding theorems hold for a given number n', in particular, 

 errors relating to the factors A+r { B of A n +B n . He promised to discuss 

 later the objections which can be raised against proofs in his preceding 

 paper. 



Cauchy 56 further developed the subject and admitted at the end of his 

 final paper that his 54 basal theorem is false, failing for n = 23. 



Cauchy 57 obtained results most of which are included in Kummer's 

 general theory. In the fifth paper, p. 181 (Oeuvres, p. 364), he stated that 

 a n +6"+c n = is impossible in relatively prime integers not divisible by 



47 Jour, de Math., 12, 1847, 137-171, 172-184. 



48 Nouv. Ann. Math., 6, 1847, 132-4. 



49 Comptes Rendus Paris, 24, 1847, 899-900; Jour, de Math., 12, 1847, 136. 



60 Comptes Rendus Paris, 24, 1847, 430-4. 



61 Comm. Soc. Sc. Gotting. Recentiores, 7, 1832, 46; Werke, II, 1863, 117. German transl. 



by H. Maser, Gauss' Untersuchungen iiber hohere Arith., 1889, 556. 



62 Comptes Rendus Paris, 24, 1847, 469-481; Oeuvres, (1), X, 240-254. 



63 Ibid., 347-8; Oeuvres, (1), X, 224-6. 



" Ibid., 516-528; Oeuvres, (1), X, 254-268. 



K Ibid., 578-584; Oeuvres, (1), X, 268-275. 



66 Ibid., 633-6, 661-6, 996-9, 1022-30; Oeuvres, (1), X, 276-285, 296-308. 



57 Ibid., 25, 1847, 37, 46, 93, 132, 177, 242, 285; Oeuvres, (1), X, 324-351, 354-371. 



