CtJRIOUS NUMERICilL PROPOSITION DEMONSTRATED* igg 



that is, each of the coefficients, except those of the first 

 and last term, js of the fprrn ji e, and is therefore divisible 

 by n, 



Q.E. D. 



Cor. We may therefore in all cases, where brevity re- Corollary. 

 quires it, write 



(P jr^ qY—p^ + w;)"*'^ + n a p'^'^q^ -\- n hf^-^q^ &c, 

 + »ijp 5"-' + 5", n being an integer prime number. 



Proposition 1, 



If the equation x" — y" = 2" be possible, then one of the Proposition H. 

 four following conditions must obtain ; viz., 



r X — y ~ r"" Cx — y — tf-W 



1st. ^ ar — • 2 =: *" 2^- <x — z := a-" 



(y + z - e (y + % = f" 



Cy:-^y — 1-^ r X — y rr r" 



3i!. -J X -^ 2 — n"*'5" 4th, ^ X — z =5" 



(y + 2=:r ly + z =«— T 



where r, s, and <, may represent any numbers whatever, 

 indicating only, that (x — y), [x — z), (y -{- z), &c., are 

 complete ?ith powers, or that they are of the form r", 5% t", 

 &c. : which proposition [ first demonstrated in the 2d edi- 

 tion of the English translation of Euler's Algebra, 



Now from what we have before observed, x, y, and z, Demonstra? 

 may be considered as prime to each other, and sipce x > y, ''O"- 

 make ^ — y + »'» then since x is prime to y, r must also 

 be prime both to x and y; for if y and r had a common 

 measure, x must have th^ same, because x z:z y -{• r, and 

 if r and x had a common ipeasure, y must have the same, 

 since x -— r zz y; therefore, as x is prime to y, r is prime 

 to bqth :p and y, Novv substituting; for x oyr e(|uatioi;^ 

 becomes 



(y + r)" — y"-2% 



from the developement of which, and substituting for the 

 coefficient of (y + r)" 



], w, n a, n 6, &c.s » a, w, 1 (Cor. Lem. 3), we abtain 



«3? 



