CURIOUS NUMERICAL PROPOSITION DEMONSTRATEO. ^OJ 



^ — n s^'-^y ■\- na s^'-^y^ — nb s^-^y^ + &c. nsy^'-^zzx^ 



or 



(^n.i _ j^ s^-^^y-l-n a s^'^^y^ —nb s'^^^y'' + &g. n y"-') s—x" 



And here, from the same reasoning as that employed above, 

 we find that 5 must be of one of the forms 9", or «"~*(p'". 

 Hence then if the equation 



x" — 3^" r: 2" 

 be possible, the following conditions must obtain, viz. 

 The difference of the roots x — y, of the form r" or n""^ r*. 

 The difference of the roots x — z, of the form s^ or w"~^ *". 

 The fum of the roots y-{-z, of the form <" or n"-* t\ 



But since (jr — y] or (x — z) and {y-{-z) are respectively di- 

 visors of the three «th powers, 2°, y", and a.", and since these 

 three quantities are prime to each other, their divisors 

 must also be prime to each other, and consequently only 

 one of these can be of the latter form above given, as they 

 would otherwise have a common divisor w. Tlierefore, if 

 the equation be possible, we shall have either 



a: — y of the form r" 



X — z s^ 



y+z t" 



or, two of these quantities will be of this form, and the third 

 of the form n"-*(p" which evidently resolves into the four 

 following cases, one of which must necessarily obtain if the 

 equation x" — y"—z^ be possible, viz. 



{X — y :z: r* ( x — y zz n*~ ' r" 



.r — z r= s* 2nd. <! x — z zz s^ 



y -\- I — i^ [. y ■\- z — t* 



{X — y n »•" Tx — y zz y° 



X — z zz w"-'*" 4th. 4 X — z = 5" 



^y ->r z zz t^ \y \- z — n"-'i" 



Q. E. D, 



Proposition 2. 



The equation x"— -r/^nz" is impossible in integers, n be- Pfop, 2. 

 ing any prime number greater than 2. 



We have before observed that x, y, and z, may be consi- Oemonstra- 

 dered as prime to each otiier, and by the foregoing prop, it tion. 



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