200 CURlaUS NUMERICAL PROPOSITION DEMON STJRATED. 



n y"-V + n a y^-^r^ -{- n b ^"-'r^, &c., n 3/ r"-' + r" = r", 

 or 



(nif""* + nay^'-^r -\- k Jy-'r', &c., n j/r"-3 + r"--'') rrr^". 

 And here it is evident, that the first side of the equation is 

 divisible by r, once, and after that, neither by r, nor by 

 any factor of r, except n be one of its factors, in which case 

 the first side is divisible by n r, once, and after that, neither 

 by n, nor r, nor by any factor of r; because the first term 

 3/"7' has no common measure with r, but all the rest of the 

 terms have ; and, therefore, the whole quantity, taken col- 

 lectively, has no common measure with r. And the same 

 inu&t necessarily be true of the equal quantity s", namely, 

 that it is divisible by r, once, and after that, neither by »-, 

 nor by any factor of r, unless n be one of its factors, in 

 which case it is divisible by n r, once, and after that, neither 

 by n, nor r, nor by any factor of r ; therefore 7; in the first 

 case, and n r, in the second case, must be a complete nth 

 power (Lemma 2) : but if r w == <?»", n must be a factor of 

 9, that is <pzzn p', therefore, r n zz w"^'", or r ~ n"-*(p'", 

 that is r ~ X — y must be of one of the former, 9", or 



And it is evident that we should have been led to the 

 same result, if we had considered the equation under the 

 form • 



x" — 2" zr i/% 



namely, that x — 2; must also be of one of the forms 9" 

 or m"*'9". 



If we investigate the same equation under the form 

 ?/° + 3" rr .r", and make y 4- 2; — 5, or s — ^ — 2> we shall 

 iind that 5 is prime to both y and z, for if s and y had a 

 common measure, z must have the same, and if s and z 

 had a common measure, then y must have the same, be- 

 cause s — z zzy, since, therefore, y and z are prime to 

 each other, s is also prime to both. Substituting there- 

 fore for z as above, our equation becomes 



{s — y) "+ y^ =: X" 



Or by expanding the binomial {s — y)" and substituting 

 for the coefficients as before, we obtain 



a" 



