CHAP, xxvi] FERMAT'S LAST THEOREM. 743 



H. Wronski 66 pretended that the impossibility of x n +y n = z n , n>2, 

 follows from his 67 results on z n Nv n = Mu n . 



F. Landry 68 proved Legendre's 17 statement for p = mn+l, w = 10 and 

 14, when n>3, noting that (14 7 1)/(141) are primes. 



Landry 69 employed two primes and 6 = 2t<f>+l, and an integer 

 belonging to the exponent modulo 6. The congruence l + e x e v =Q 

 (mod 6) can be reduced to 1 -f- e d= t z = unless x = <j> or x = 0, whence 20 = 1 . 

 By use of the substitutions e = e7 1 , e=e} /z , etc., we can reduce l + e+e z =0 

 to a similar congruence with z replaced by the integral residues modulo of 



1 z-l 1 z 



Z, 12, , 



z' z 1-2' z-l' 



Excluding 2 = 1 or 2, these six expressions are incongruent modulo <j> unless 

 <l> is of the form 6Z+1 and then they reduce to two for two special values 

 of z. If all three relations 1 + e e z =0, 1 e+e z =0, 1 e Z =Q are im- 

 possible for a single one of the above six values, then 1 + e e z =0 is im- 

 possible for all six. 



For Landry 's third memoir (on primitive roots), see Vol. I, p. 119, 

 p. 190 of this History; for his fifth memoir (on continued fractions), see 

 Landry 69 of Ch. XX above. 



Landry 70 recurred to the exception arising if 2*=1 (mod 0), where 6 

 is a prime 2k<j>n+l, n a prime >2. For = 5, 7, 11, 13, 17, 19, he found 

 all the cases in which 2 0I Fl has such a factor 6. For example, if = 11, 

 only when n = 31, = 683. Aside from these exceptions, l + e 2 =0 does 

 not hold for 2 = or 2 = when 0^19; nor for 2 = 2, , 1, or 2 = 3, 13, 

 I, etc., except for a few special values of 6. 



Landry 71 proved that, if 8 is a prime 2&0n+l(n>3), l + ee z =0 

 (mod 6) are each impossible for = 5, 7, 11, 13, 17, 19, aside from the excep- 

 tions for = 11, 13, 17 noted by Landry, 70 and the new exceptions, aris- 

 ing for = 19: = 761, n = 5, & = 4; = 647, n = 17, k = l; = 419, n = ll, 

 fc=l. 



H. E. Heine 710 considered P m DQ m = l, where P, Q, D are polynomials 

 in x. 



L. Calzolari 72 noted that any given numbers x, y, z can be expressed in 

 the form x = v+w, y = u-\-w, z=u-\-v-\-w [since we may take u=zx, 



66 Veritable science nautique des marees, Paris, 1853, 23. Quoted in I'interme'diaire des math., 



23, 1916, 231-4, and by Guimaraes. 273 



67 RSforme du savoir humain, 1847, 242. See p. 210 of Vol. 1 of this History. 



68 Premier me'moire sur la theorie des nombres. Demonstration d'un principe de Legendre 



relatif au theoreme de Fermat, Paris, Feb. 1853, 10 pp. 



69 Deuxieme me'morie sur la the'orie des nombres. The'oreme de Fermat, Paris, Julj r , 1853, 



16pp. 



70 Quatrieme me'morie sur la theorie des nombres. The'oreme de Fermat, Paris, Feb. 1855, 



27pp. 



71 Sixieme memorie sur la theorie des nombres. Theoreme de Fermat, 3 e livre, Paris, Nov. 



1856, 24 pp. 

 710 Jour, fur Math., 48, 1854, 256-9. 



72 Tentative per dimostrare il teorema di Fermat . . . x n +y n =z n , Ferrara, 1855. Extract 



by D. Gambioli, 171 158-161. 



