. Carnegie Institution of Washington publication. CHAP, xxvi] FERMAT'S LAST THEOREM. 739 Then zn-yn = xn Then Zfzm and Y/ym give. 2znl2, 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. Jacobi40 gave a table of the values of m' for which l-\-gm==gm' (mod p), where p is a prime ^103, 0 ^ra^!02, and q is a primitive root of p. 0. Ter quern41 proved the theorem of Lebesgue31 and the corollary of Liouv
. Carnegie Institution of Washington publication. CHAP, xxvi] FERMAT'S LAST THEOREM. 739 Then zn-yn = xn Then Zfzm and Y/ym give. 2znl2, 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. Jacobi40 gave a table of the values of m' for which l-\-gm==gm' (mod p), where p is a prime ^103, 0 ^ra^!02, and q is a primitive root of p. 0. Ter quern41 proved the theorem of Lebesgue31 and the corollary of Liouville32. A. Vachette42 noted that xm—yn = (xy}p is impossible in integers. For p = mn, set z = (xy)n and take n = m. Thus xm—ym = zm is impossible if z is a power of xy. J. Mention43 proved the formula [cf. Kummer25]: (6) a V. A. Lebesgue44 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, xn+yn=zn 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. Liouville46 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,
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