542 



Sir F. Pollock on Fermafs Theorem of 



[Dec. 15, 



III. On Fermat^s Theorem of the Polygonal Numbers^ with Supple- 

 ment. By the Right Hon. Sir Frederick Pollock, F.R.S. 

 Eeceived December 5, 1864. 



(Abstract.) 



This paper (with its Supplement) proposes a proof of the first two 

 theorems of Fermat, relating to the polygonal numbers, viz. that every 

 number is composed of not exceeding three triangular numbers, and not 

 exceeding four square numbers. And this is done by a method entirely new, 

 founded on the properties of the triangular numbers and the square 

 numbers, and the relation they bear to each other, and on the expansion 

 of an algebraical expression of three members into a lincj a square, and a 

 cube, so as to obtain every possible value of the whole expression ; and 

 throughout the proof every number or term in a series (except in the Table) 

 is expressed by the roots of the squares that compose it, and the roots 

 only are dealt with, and not the numbers or the squares that compose them ; 

 a Table is constructed from the triangular numbers, thus (see opposite 

 page). 



Mode of constructing the Table. 



The series of triangular numbers is in the centre of the Table. Below 

 that series the adjoining terms are united, and they form the square numbers 

 1, 4, 9, &c. ; the next adjoining terms are united, and they form the next 

 row, and so on. 



Above the triangular numbers each term is doubled, forming the series 

 above, and then the adjacent terms are added together and form the next 

 row, and so on ; the differences above are 1, 3, 5, 7, 9, &c. (2n-\- 1), and 

 below are 2, 4, fi, 8, 10, &c. {2n). 



From the examination of which Table it appears that the sum of any 

 two triangular numbers, however remote from each other in the series, is 

 always a square number plus a double triangular number ; that is, 



and the difference between the sum of two triangular numbers and the sum 

 of some other two triangular numbers may be any number whatever, odd 

 or even, positive or negative. The first of these propositions is mentioned 

 and proved algebraically in the Philosophical Transactions of the year 

 1861, p. 412 ; the result is perhaps more clear when presented in a tabular 

 form as above (but more extensively and at large in the paper) ; it is ob- 

 viously capable of strict algebraical proof. These two propositions, and the 

 expansion of an algebraical expression into a line, a square, and a cube 

 (exhausting every possible value of the expression), are the foundation of 

 the whole proof, which, in addition to proving the first and second theorems 

 of Fermat, proves also that every odd number has in some form or other 



