]^6 CVRtOtJS NUMERICAL PftOf O^tttolf tifeMONStRAttet 



and in which latt&r equation the three resulting quantities, 

 Xy y, and .'i, are prime to each other; and, conversely, if 

 the latter be impossible, the former is impossible also; we 

 shall therefore only consider the case in which x, y, and r, 

 s are prime amongst themselves* 



3. It will be sufficient to consider the ambiguous sign + 

 under either of its forms + or — -: for if the equation x^\- 

 ^"rrz" be possible, so also is the equation £" — y" rr of" ; 

 and if the equation be impossible under the latter form, it 

 is likewise impossible under the former. 



We shall therefore limit our demonstration to the equa- 

 tion a;" — y ~ z"> in which n is a prime ntimber, and a;, y, 

 and z, numbers prime to each other: the impossibility of 

 which, from what Is shown above, involves with it the im-^ 

 possibility of the general equation vC" + 3/" rr 2", when .r, yi 

 and 2, are any numbers whatever, and n any number ex* 

 cept 2, or some power of 2. Now with regard to « rr 2, 

 we know, that the equation is not impossible, and the case 

 of n =:4 has been demonstrated to be impossible by Euler^ 

 and others; and this latter case involves that of every higher 

 power of 2, thus x' + ?/» - 2« - (x*)* + (/)♦= (2*)*; which 

 being impossible in the latter form, is necessarily so in the 

 former: and in the same manner, the impossibility of the 

 equation for any higher power of 2, may be shown to ber 

 involved in that of n = 4* it is evident, therefore, that our 

 equation, together with that of » "": 4, involves every possi-* 

 ble value of n greater thaa 2. 



Lemma 1. 



h 



i rraiiia 1. If there be two fractions, as — , and -- , each in its 1ovt« 



est tefmsj and of which the denominator of the one con-« 

 tains any factor not common with the denominator of the 

 ©ther; then I say, that neither the sum nor difference of 

 those fractions can be equal to a complete integer num- 

 ber. 



Let — and — be any two fractions in their lowest terras, 

 A B -^ 



«ioi that a is prime to A, and h prime to B, also suppose B 



ta 



