234 report— 1859. 



It may be considered from two different (though closely connected) points 

 of view, each of which has proved equally fertile in consequences. First, it 

 may be regarded as asserting that, if p be a prime number, and x any num- 

 ber prime to p, the remainder left by the power xP~ l when divided by p is 

 unity. It is thus the fundamental proposition in the arithmetical theory of 

 the residues of powers, or, which is the same thing, of binomial congruences. 

 Or, secondly, it may be regarded as asserting that the congruence x p ~ l = 1, 

 mod p, has precisely p—\ roots; and that these roots are the terms of a 

 system of residues prime to p. It is in this latter point of view that the the- 

 orem is the basis of the general theory of congruences. 



We may observe that the demonstrations of Fermat's Theorem point to this 

 twofold aspect. 



The proof which is found in most English treatises of Algebra (it is the 

 first of those given by Euler*), and which depends on the property of the 

 binomial or multinomial coefficient, would naturally lead us to regard the 

 Theorem in the first point of view. The same may be said of Euler's second 

 demonstration-)-, which consists in showing that the index of the lowest power 

 of # in the series 1, x, x 2 , x 9 , &c, which leaves unity for its remainder when 

 divided by p, is either p — 1, or some submultiple of p — 1 ; or again of the 

 demonstration of MM. DirichletJ, Binet§, and Poinsot||, which depends on 

 the observation that the terms of a system of residues prime to any modulus, 

 being multiplied by any residue prime to the modulus, still form a system of 

 residues prime to the modulus. 



But a remarkable proof of the theorem, in the second expression we have 

 given to it, occurs in a memoir of Lagrange^]". As this proof (though very 

 elementary) has not been copied by subsequent writers, and is consequently 

 but little known, its nature may be indicate d here. 



Let the product 



(x+l)(x+2)(x + 3) (x+p-1) 



be represented by 



xr-i + A 1 xP-2 + A^ a??-3 + . . . . A p _ 2 x + A p - U 



x denoting an absolutely indeterminate quantity. Writing # + 1 for x, and 

 multiplying by # + 1, we obtain the identity 



(x+iy + A l (x + \)p-i + A 2 (x+l)p-2+ ... +A p _ l (x+1) 



= (x+p)[xP-i + A 1 xP-2 + A.,xP- 3 + ..+A p - 2 x + A p _ 1 ]; 



whence, by equating the coefficients of like powers of x, we find, 



A p(p-l) 

 1 J. '2 



* Comment. Acad. Petropol. vol. viii. p. 141, or Comment. Arith. vol. i. p. 21. This is 

 the first demonstration of the Theorem discovered, since the time of Fermat. The memoir 

 containing it was presented to the Academy of St. Petershurgh, Aug. 2, 1 736. 



t Novi Commentarii Petropol. vol. vii. p. 49, or Comment. Arith. vol. i. p. 260. From 

 the point of view in which Fermat presents his theorem, it is not improbable that the de- 

 monstration he had found of it was no other than this of Euler's. (See Fermati Opera 

 Mathematica, Tolosae, 1679, p. 163.) It has been adopted by Gauss in the Disquisitiones, 

 Art. 49. 



t Crelle's Journal, vol. iii. p. 390. 



§ Journal de l'Ecole Polytechnique, Cahier xx. p. 289. 



|| Reflexions sur la Theorie des Nombres, p. 32. But the principle of this demonstra- 

 tion is employed by Gauss in a memoir published in the Comm. Soc. Gotting. vol. xvi. 

 p. 69, to which we shall have again to refer. (See Art. 19 of this Report.) 



T[ Demonstration d'un Theoreme nouveau concernant les Nombres Premiers (Nouveaux 

 Memoires de l'Acade'mie Royale de Berlin, 1771, p. 125). The 'new theorem' is that 

 known as Sir J. Wilson's. 



