JOURNAL 



OF 



NATURAL PHILOSOPHY, CHFMISTRY, 



AND 



T II E ARTS. 



JANUARY, 1802. 



ARTICLE I. 



On the Exhibition of a Series of Pri??ies> and the Refolution of a 

 Compound Number into all its Fdclors. By Mr, J. Gough. 



To Mr. NICHOLSON. 



SIR, 



EULER remarks in his Algebra, that mathematicians Natural feries of 

 have not yet difcovcred a way of exhibiting a natural feries of ["^nof^num" 

 primes; and in reality I know of no writer on arithmetic, bcr into its fao 

 who fhews how a compound number may be refolved into all *°" " ot >' ct ef ~ 

 its factors, other wife than by trial. The following method, 

 however, may be faid to folve both problems, perhaps in as 

 ready a way as the nature of numbers will admit ; and on fixed 

 principles : it depends upon the following property of com- 

 pound numbers. 



If a compound number, which is not a fquare, be refolved Property of 

 into any two of its fadors, one of thefe fadors will be lefs than |;°™? ound ^^ 

 the root of the next greater fquare, and the other will be equal 

 to, or greater than the fame root. — If you deny the property, 

 then both factors mult be either greater than the root of the 

 fquare, in which cafe the product will be greater than the 

 fquare ; or, they muft be both lefs, than that root, on which 



Vol, I.— -January. B fuppofition 



