418 SIR F. POLLOCK ON FERMAT’S THEOREM OF THE POLYGONAL NUMBERS. 
Series. 
Roots. 
Numbers. 
Roots. 
Numbers. 
19= 
1,1 
® 
also = 
0, 0 
® 
23= 
2,2 
= 
1,1 
® 
31 = 
3,3 
2,2 
43= 
4,4 
3, 3 
® 
59 = 
6,5 
4,4 
@ 
79 = 
6, 6 
= 
5,5 
@ 
103= 
7,7 
® 
zz: 
6, 6 
@ 
131 = 
8,8 
@ 
= 
7,7 
0 
163= 
9, 9 
8,8 
® 
The numbers are alternately of the form and in — 1; the terms of the series 
are therefore equal to 2 squares + a number of the form 4w+l, and to 2 other squares 
+ a number of the form in—\. A number of the form in—\ cannot be composed of 
less than 3 squares; for if and ¥ be odd squares, their sum is of the form (8w-f-2); if 
even squares, of the form {in); if one be odd and the other even, and in — 1 
cannot equal 8?i'+2, or 4w", or 4^"'4-l; but as the 2 squares are always equal, the 
arithmetic number may always be turned into a number of the form in-\-l, by substi- 
tilting for the 2 equal squares 2 others, whose roots shall be, the one one more, the 
other one less ; thus 79—6, 6 + {^ =5, 7 +5 = 5, 7, 1, 2 ; also=5, 6+(2^ =5, 5, 5, 2 , 
And eveiy term of the series is divisible into 4 squares whenever in-\-\ is divisible into 
2 squares, or when in! — 1 — 2, another form of 4w+l, is so divisible. It would follow, 
that if there be any 2 series in arithmetical progression with a common difference of 1, 
and the odd terms of the one be placed over the even terms of the other, then if either 
series be considered as composed of roots and the other of numbers, and the squares of 
^ the roots be added to the numbers, a series will be formed of the first sort ; thus 
9, 8, 7, 6, &c. 
6, 7, 8, 9, &c. 
If the lower be considered as roots, the series becomes 
45,12 b7, ,4 71,16 87,13 &c. ; 
if the upper be considered as roots, the series is 
87,16 71,14 57,12 45, &c.. 
