( 45 ) 



V. — On the Theory of Numbers. By H. F. Talbot, Esq. 



(Read 21st April 1862.) 



The object of this paper will be, to give a connected view of some theorems 

 of importance, which are often found in books rather obscurely demonstrated, 

 and in some cases are inaccurately given, or are liable to exceptions which are 

 not mentioned. 



§ 1. On FermaVs Theorem, and Wilson's Theorem. 



The most convenient starting-point for this investigation seems to be the 

 well-known theorem, "If j9 is a prime number, and (,2? + l)^ is expanded by the 

 binomial theorem, all the coeflflcients, except the first and last, are divisible by p. 



For it is obvious, in the first place, that all the coefficients are integers. If 

 we multiply x+l into itself, any number of successive times, the coefficients arise 

 from the multiplication and addition of integers, and are therefore themselves 

 integers. 



Next, the binomial theorem gives the coefficients in the form 



p(j>-V) p(p-l)(p-2) 

 p, 2 ' 2-3 ' 



Let us consider any one of these, for instance the last; then, since 

 ^ 2-s ^^ ^^ integer, the numbers 2 and 3, found in the denominator must 



divide some of the factors in the numerator. But they cannot divide p, it being 

 a prime by hypothesis; consequently, they divide (p—l) (p-2), therefore 



L^~ is ^^ integer. But this integer is the quotient of the coefficient 



divided by p. Therefore, p divides this coefficient, and so for all the others. 



This is the place to introduce a convenient notation, invented, I believe, by 

 Gauss. 



If a and b are two numbers which, when divided by the number n, leave the 

 same remainder, Gauss says that they are congruous to each other, according to 

 the modulus n ; which he expresses thus, a^h (mod. n). The sign ^ is imitated 

 from ^= the sign of equality, and implies, not that the numbers are really equal, 

 but that they are equivalent (under certain circumstances only). For, if a^h 

 (mod. n) this would not, in general be the case with a different modulus. 



I propose in the present paper sometimes to use the word equivalent instead 

 of congruous. 



VOL. xxin. part I. N 



