412 SIE F. POLLOCK ON FEEMAT’S THEOEEM OF THE POLYGONAL NUIkLBEES. 
5 , 0 , 2 
— 4, 1, 1, 3 when squared and added . =27 
2 , 1 , 5 
— 3, —1, 0, 5 whose squares .... =35 
The proof of all this depends on a property of numbers mentioned in the Philosophical 
Transactions for 1854, vol. cxliv. p 317. 
If any number be composed of two triangular numbers, it will also equal a square and 
a double triangular number. If 
■r 2 ’ 
it uill be of the form and may be assumed equal to For if 2 
numbers be both odd or both even, they may always be represented by a-{-h and a—h’, 
if one be odd and the other even, they may always be represented by 
a — J, or by and a — Z»+l ; and if the two numbers be made the bases of trigonal 
numbers, the sum of the two trigonal numbers will always be of the form or 
now when any number in the natural series of numbers is composed of two 
triangular numbers, it may be represented by and Iw + l will then equal 
4«'^+4«4-14-45^ — obviously the sum of an odd and an even square, whose roots ai’e 
25 and 2a+l ; and 2?^+l, the corresponding odd number, will equal 
— obriously composed of 4 square numbers, whose roots are 5, 5, «, a-\-l ; and if they 
be arranged thus, 
25, — (a+5) 2a-{-l 
—5, 5, —a, a+l? 
so that the sum of their roots may equal 1, the exterior diiferences of the roots will be 
25 and 2a+l, the roots of the two squares into which 4w.+l is divisible ; and the middle 
difference will be — (a-f 5), the smaller half of the sum of the roots (25+2(2+1) with a 
negative sign ; if the exterior differences be reversed and the middle difference be 
increased by 1, the differences will be 2a+l, — (a+5 — 1), 25, and the roots whose sum 
will equal 1 will be, with their differences above them. 
2«+l, _(«+5-l), 25 
-(«+l), a- (5-1), 5+1, 
and the sum of the squares of the roots will be 2 more ; from these two sets of roots all 
the rest may be obtained, by adding one to each of two roots and subtracting 1 from 
each of the other two roots ; the exterior differences of the roots will therefore always 
be the same, and the middle difference will increase by 1 at each step ; the sum of the 
squares of the roots will increase by 
• 2, 4, 6, 8, &c. 
As the sum of any two square numbers of which one is odd and the other even 
(4«^+ 4(2+ 1 + 45®) must be of the form 4n-{-l, every possible case of an odd square 
