CHAP, xxvi] FERMAT'S LAST THEOREM. 739 



Then 



z n -y n = x n 



Then Zfz m and Y/y m give 



2z nl2 , 

 From the sum and difference of the resulting values of 



qly 



Developing the difference of the two members by the binomial theorem, 

 we get a series hi y/z with every coefficient negative if n>l. Next, the 

 case n = 2m is treated at length. 



C. G. J. Jacobi 40 gave a table of the values of m' for which l-\-g m ==g m ' 

 (mod p), where p is a prime ^103, ^ra^!02, and q is a primitive root of p. 



0. Ter quern 41 proved the theorem of Lebesgue 31 and the corollary of 

 Liouville 32 . 



A. Vachette 42 noted that x m y n = (xy} p is impossible in integers. For 

 p = mn, set z = (xy) n and take n = m. Thus x m y m = z m is impossible if z 

 is a power of xy. 



J. Mention 43 proved the formula [cf. Kummer 25 ]: 



(6) a 



V. A. Lebesgue 44 obtained (6) by applying Waring's formula to the 

 quadratic equation with roots a, b. Applying it to the cubic with the roots 

 a, /3, y, we get (a+/3+7) n . For n = 7, the latter result is said to have been 

 employed [in papers 28-SCf] to prove the impossibility of x ! -\-y ! =z ! by a 

 method simpler than that for exponents 3 and 5. 



G. Lame" 45 claimed to have proved that, if n is an odd prime, x n +y n =z n 

 is not satisfied by complex integers 



(7) ao+air+---+an-ir' 1 - 1 , 



where r is an imaginary nth root of unity and the <z's are integers. 



J. Liouville 46 pointed out the lacuna in Lamp's proof that he had not 

 shown that a complex integer is decomposable into complex primes in a 

 single manner. 



Lame" (p. 352) admitted the lacuna and believed (on the basis of exten- 

 sive tables of factorizations) that it could be filled; he affirmed (pp. 569-572) 

 that the ordinary laws for integers hold for complex integers when n = 5. 



40 Jour, fur Math., 30, 1846, 181-2; Werke, VI, 272-4. 



41 Nouv. Ann. Math., 5, 1846, 70-73. 

 Ibid., 68-70. 



43 Nouv. Ann. Math., 6, 1847, 399 (proposed, 2, 1843, 327; 18, 1859, 172, 249). 



44 Ibid., 427-431. 



Comptes Rendus Paris, 24, 1847, 310-5. 

 46 Ibid., 315-6. 



