ANNALS 



OF 



PHILOSOPHY. 



FEBRUARY, 1826. 



Article I. 



Extension of a Theorem of Fermat. By Mr. W. G. Horner. 

 (To the Editors of the Annals of Philosophy.) 



GENTLEMEN, Dec. 26, 1825. 



A PROPERTY oi prime numbers, which was first stated by 

 Fermat, and which serves so important uses in the analytical 

 division of the circle, and is the basis of the theory of decimal 

 circles, has been demonstrated in two very different ways. 

 The process of Euler is familiar to all, having been repeated by 

 subsequent writers on the theory of numbers. But the manner 

 adopted by Mr. Ivory, in vol. i. of the Math. Repos. possesses 

 important advantages in regard of conciseness, and of the readi- 

 ness with which the train of reasoning may be extended to 

 tiumbers in general. This circumstance occurred to me some 

 eighteen or twenty months ago, when pursuing a little specula- 

 tion on decimal circles ; the properties of which may be deve- 

 loped to their utmost limit by the aid of this discovery. Grate- 

 ful to the source of my success, I immediately addressed a copy 

 of my extension of the basis which he had supplied to Mr. 

 Ivory, but am uncertain if it reached its destination. That 

 gentleman can, however, have no objection to its appearance in 

 your respectable work; and being unwilling, not for my own 

 sake only, that my theorem should remain in silence, I shall 

 feel obliged by your early inserting it. 



I am, yours very truly, 



W. G. Horner. 



Theorem. — If P, p, be prime to each other, and n indicate the 

 number of integers less than p and prime to it, P" — 1 will be 

 divisible by p. 



New Series, VOL. xi. g 



