SIR F. POLLOCK ON CERTAIN PROPERTIES OF SQUARE NUMBERS, ETC. 317 
If the following series of equations be assumed, 
1+2 d =2</ + l 
3+2c?, = 2g' + 1 
7 +2+=2</+ 1 
13 + 2+=2^+l 
w 2 +m+l+2f/ m _ 1 = 2^+l, 
and if each of the quantities 2c?+l, 2^+1, 2<4+L 2++1 ...2d m _ x -\- \ can be resolved 
into 4 square numbers, the algebraic sum of whose roots=l, then the given odd 
number 2^+1 may be resolved into successive sets of 4 squares, the sum of whose 
roots will be successively 1, 3, 5, 7---2m+l. Hence an odd number, 2q-\-\, may be 
resolved into 4 square numbers, the sum of whose roots shall be equal to 2j»+l, if 
upon adding 1 to the difference between 2^ + 1 and the (p+l)“ term of the gradation- 
series, the difference so increased can be resolved into 4 square numbers, the alge- 
braic sum of whose roots=l. If it be required to resolve 37 into 4 squares, the sum 
of whose roots shall equal 7=2x3 + l, here p=3, the (j»+l)“or 4th term of the 
gradation-series is 13; 13 is of the form 4‘2 2 — 2’2+l, and it equals 2 2 +2 2 +2 2 +l 2 
(2— l) 2 ; the difference between 37 and 14 = 24, increased by 1=25, 25 = l 2 +2 2 +2 2 
+4 2 , and the roots+ 1 — 2 — 2 + 4= 1 and 13 + 24= (2 + 2) 2 = 4 2 , 4 2 , l 2 , — 2 2 
(2 + 2) 2 
(2 — l ) 2 
(2 — 4) 2 
and +4+4+1— 2 = 7- 
If, therefore, every odd number can be resolved into integral square numbers (not 
exceeding 4) whose algebraic sum wiil equal 1, then every odd number can be 
resolved into integral square numbers (not exceeding 4) whose algebraic sum will be 
], 3, 5, &c. [viz. all the odd numbers up to the maximum ]. 
I propose (in order not to leave the Theorem C. unproved) to show by the proper- 
ties of numbers already proved, that every odd number may be resolved into integral 
square numbers (not exceeding- 4) whose algebraic sum will equal 1. 
Every odd number may be represented by 2p-\-\ (p being any integer) : then by 
Fermat’s theorem of the polygonal numbers (as proved by Legendre, Theorie des 
nombres),7? must either be a trigonal number, or composed of two or three trigonal 
numbers. If it be a trigonal number, then p — — g— , and 2p-\-\ = ^ 2 + </ + 1, which 
equals 4w 2 +2« + l, which is divisible into (w+l) 2 ,?z 2 , w 2 , w 2 , and n— w+w.+ (w+l) = l. 
If p be composed of 2 trigonal numbers, p = v , and the sum of any two 
trigonal numbers is of the form of a 2 + u+6 2 * and may be assumed equal to <r+a + & 2 , 
* If 2 numbers be both odd or both even, they may always be represented by a + 6 and a— b ; if one be odd 
and the other even, they may always be represented by a + i + 1, a — b or a + b, a — 6+1; and if the 2 numbers 
be made the bases of trigonal numbers, the sum of the 2 trigonal numbers will always be of the form a 3 + a 
+ 6 2 or a 2 + 6 + 6;. 
