*T|()^ CURIOUS NUMERICAL PROPOSITION BEM0K8TBATED. 



by A; but by the supposition the quotient is not again divi- 

 sible, either by 7-, or by any factor of r, therefore r must 

 in this case be equal to A". 



Again, if A be not a prime number; let it be resolved 

 into its prime factors, as A zz a b c, &c., or A." zz a" i" c", 

 &c. Now if A" r: a" 6" c", &c., be divided by any power of 

 a less than a", the quotient will be again divisible by a ; 

 and in the same manner if it be divided by any power of 6 

 less than 6", the quotient will still be divisible by 6, and so 

 on of any other divisor that is not a complete nth power; 

 and, therefore, conversely, if A" be divisit^le by any other 

 number r, once, and after that, neither by r, nor by any 

 factor of r, then must r itself be a complete nth power. 



Q. E.D. 



Lemma 3. 

 iemmaS. In the expanded form of the binomial (p -\-qY, when 



n is a prime number,' each of the coefficients, except those 

 of the first and last terms, are divisible by n. 

 For each of the coefficients is of the form 



n . (n — 1) . (n — 2) &c. (n — r) 

 ""1 '. i '. 3 kc. r + 1 



which is always an integer, from the nature of the bino« 

 mial ; and since w is a prime number it is not divisible by 

 any of the factors in the denominator, which are all less 

 than Jt, except in the coefficient of the last term, which 

 does not enter into our consideration, the coefficients of the 

 first and last term being excepted in the proposition. 

 Since then 



n . {n — -. 1) . (n — 2) &c. n -— r 

 1.2.3 &c. r + 1 



is an integer, and n is prime to all the factors in the de- 

 nominator, therefore, 



(w — 1) . {n — 2) &c. jn^r) '_ 

 2 . 3 &;c. r + 1 — ^ 



Is also an integer, and consequently 



n . [n — 1) . (m >— 2) &c. {n — r) __ 

 1 . 2 . 3 &c. (r + l)-'^^*' 



that 



