46 MR H. F. TALBOT ON THE THEORY OF NUMBERS. 



If any number a is divisible by n, it is equivalent to zero, with modulus w, 

 which is written a^o (mod. n). 

 To return to the last theorem. 

 If/> is a prime, 



{x + iy = {ocP + l)+ (p. ,t;P-i + ^^^~ ' a;P-^ + &c.\ 



Therefore we have the congruence or equivalence, 



{a; + ly^ 00^ + 1 (mod. p). 



For all the other terms vanish, their coefficients being all divisible by 7^, whence, 



p=0(mod.^) P^^=0 1SPz1)^ = o, 



and so on. 



Take this equivalence (x + lf ^ x^ + l 



and suppose a; =1 .-. 2^^ ^ 1 + 1 ^ 2 



Next suppose x =2 .-. 3" = 2" + 1 



But we found 2" = 2 .-. 3" = 3 



Next suppose ^ = 3 ■•• 4'' ^ 3" + 1 



But we found 3^ = 3 .-. 4^ = 4 



And so on till we reach «^ ^ «• 



a being any number. Transposing, we have a^-a^O (mod. p). In other words, 

 the prime number p divides a"— a, or a («''~^-l). It therefore divides one of the 

 two factors a, or a^~^~l, whence we obtain Fermat's celebrated theorem, — " If 

 p issL prime number, which does not divide a, it necessarily divides «^~'-l." 



Next let us consider a beautiful theorem first given by Lagrange. Ifp is any 

 prime number, and an equation be formed of p-l dimensions, whose roots are 

 the series of natural numbers, 1, 2, 3, . . . . (i^-l), all the coefficients of this 

 equation (except the first and last) are divisible by p. 



Example. — Let the roots be 1, 2, 3, 4, 5, 6, the equation will be 

 a;«-21 x' + no «*-735 a;^ + l62i x''-1764: a; + 720 = 

 and each coefficient except the first and last is divisible by 7. Assuming 

 Lagrange's theorem as proved, we can deduce a remarkable consequence from it. 

 Let Z be the last coefficient, it is the product of all the roots, or Z = l, 2, 3, . . . . 

 (p-1). Z is always positive, because the equation has an even number of 

 dimensions. 



Therefore the equation may be written thus : — 



But by Lagrange's theorem, 



A = (mod. p), B = 0, &c. 



