CURIOUS NUMERICAL PROPOSITION 9EU0NSTRATEB. 1Q^ 



venons tVexposer paroissent insuffisantesJ''' (Page 410, Essai 

 sur la Theorie des Nomhres.) 



This therefore being the state of the proposition, the fol- 

 lowing general demonstration, for every possible integer 

 value of n, greater than 2, will not, it is presumed, be un- 

 acceptable to the lovers of this interesting branch of ana- 

 lysis; before entering upon which, however, at length, it 

 will be proper to make a few preliminary observations on the 

 general equation, in order to render the demonstration as 

 simple and concise as possible. 



1. In demonstrating the impossibility of the equation Observations 



a" + y" ==2", it will be sufficient to consider n as a prime *^" ^"^^ general 



— ^ eqviation, 



nurtiber. For suppose n be not a prime, but equal to the 

 product of two or more prime factors, asn — pq, then the 

 equation becomes xPI + i/Pi — 2P1 =: (xP)i -f (^p)q — {zf)^, 

 being a similar equation, in which the power 5 is a prime 

 number; and, therefore, if the equation be possible when n 

 is a composite number, it is also possible for a prime power; 

 and conversely, if the equation be impossible when the 

 power is a prime, it is also impossible for every composite 

 power; we shall therefore in what follows consider » as a 

 prime number; 



2. We may always suppose x, y, and z as prime to each 

 other ; for it is evident, in the first place, that two of these 

 numbers cannot contain a common divisor, unless the third 

 contains the same. Suppose, for example, that a" and y"^ 

 contained any common divisor, as 9, and that 2" did not 

 contain the same, then, in the equation x"" ■\- y^ — 2", we 

 should have x"" jrjf divisible by (p, but the equal quantity 

 z" not divisible by it, which is absurd; and the same may 

 be shown if any other two of these Quantities are supposed 

 to have a common divisor which the third has not. And if 

 they have all three the same common divisor, as xrz.^k^ 

 y zz (py, and 2 — (pi, then the equation becomes (^'' x"" •\- 

 9"jK" zz (f>" «'', or, dividing by the greatest coinmon divisor, 

 *" H^jJ" — z''- : if, therefore, the equation x" + f/° = z" b* 

 possible, when x, y, and x, have a common divisor, it i» 

 also possible after biding divided by that common divisor, 



O 2 ' and 



