^vH CJURIOUS NUMERICAL PROPOSITION DEMOKSlftVi'^^'-. 



quently A must be a complete nth power, or A ~ A'^'i 

 that isi 



and, tberefore, 



I" — 5" _- r" ~ f 5 r A' , 

 pr^ diviiiing by trs,we have 



5r if St * 



v/hich muf?^ necessarily be an integer. But these tlv reef rac% 

 tions are each in their lowest terms, because r, 5, and f, sure 

 prime to each other, an«;l each of the denominators contains a 

 fa.ctor that is not common to the other two; they cannot 

 therefore be equal to an integer, by cor. Lenama 1, and con- 

 sequently the equation j^ impossible under the first condi- 

 tion. And in oxder to arrive at the results of the other three 

 conditions, we have only to substitute ?i°-^ r" instead of ?■% 

 gj''-^ 5" for 5"s and ?i"-' t" fo.r P, whence we draw the follow^ 

 |ng- four conclusions, 



1st 



rs ir St ~~ ^ 



4tli 



according as we assume the 1st, 2nd, 3d, or 4th, condition. 

 And in each of these expressions we ought to have A', A", 

 A'", A"", integer numbers, if the given equation were pos«s 

 sible ; but since in each of these expressions we have three 

 fractions each .in its lowest ternis, and the denominator of 

 each contains a factor not common to the other two, ther§« 

 fore by Lemma 1, and its coroliaryj they cannot produce am 

 inteijer number. 



Having shown, therefore, that, if the equation x' — ^j/"™^'" 

 ^.(iK^, l^o^sibie, one of the quantities A', A", A'", or A'"\ 



«"-' 





.jU-I 



y,n_ 



* 



rs 



t__ 



tr 



At 



- — 



rs 



^ 



ir 



«/"- 



St 



^n_r 



— 



u^~' 6-* 



~ 



r"- 



rs 



ir 



~7F 



?l^-^ t 



n»I 



tr 



— 



^r,^S 



rs 





St 



