78 



It may be observed tliat the velocity is the same as that acquired by a 

 body falliiiK tlirougli the lieight b, and is iiulependeut of tlie radial dis- 

 tance, r. Tlie time of descent is directly proportional to r; and both are 

 independent of the weights. That is, we liave the tlieorem: 



Motion ou the h'lix surface /.v equlrnlent to that on the inrliue plane, when r is 

 constant. 



A Generalization of Fermat's Theorem. 



Jacob Westluxd. 

 Consider the function 



n(A) n(A) 



<» F(», A ) = ;;'■"_,„»'■'■'+.... +„°'^')) 



-^-M- n(A1 



+(""'^'^-V....)-.... + ,-:;„^^^'^^^^^, 



where « is any algebraic integer and A any ideal in a given algebraic 

 number field, Pi, . . . P^ are tlie distinct prime factors of A, and n(A) de- 

 notes tlie norm of A. The tlieorem whicli we shall prove is that F( « , A) 

 is always divisible by A. 



For tlie case when a and A are rational integers several proofs of the 

 divisibility of F(«, A) by A have been given*. 



When A is a prime ideal the function F(«, A) reduces to n^^^) — n, 

 which, as we know, is divisible by A. 



Let us first consider the case when A = pf', where Pi is a prime ideal 

 of degree f, and pi the rational prime divisible by Pi. Then 



fs, 1(S,-1). 



F(., P^)=r^ _ „'" 

 But 



f(8l-ll 



Pi (Pit— 1) 



n = 1, mo d P^' 



and 



fai l(s,-l) 



P. Pi „ 



a = a , mod P,' 



hence 



(2) F(«, P^)e:-o, modP^ 



Now, suppose A^B.Pf- where B is any ideal not divisible bv Pi. 

 Then we can easily derive the following relation: 



f(s.-l) 



F(a\ B)-F(", B.Ph = F(/' , B), 



Dickson, Annals of Mathematics, 2d Series, Vol. 1, 1899, p. 31. 



