628 
SIR FREDERICK POLLOCK ON THE MYSTERIES OE NUMBERS 
These forms give the differences of roots set opposite to them ; 12 is the limit, as the 
roots. 
lowest number which can give a difference of 18 is 85 (0, 0, 6, 7). G.7 have a difference 
of 13. 
So with reference to the algebraic sum of the roots : 
4, 4, 4, 5 gives 1, 7, 9, and 17 
2, 2, 4, 7 gives 1, 3, 7, 11, 15 
0, 1, 6, 6 gives 1, 11, 13 
1, 2, 2, 8 gives 3, 5, 7, 9, 11, 13. 
It is remarkable that the sum of the roots being equal to every odd number is imme- 
diately connected with Fermat’s first proposition of the triangular numbers ; it is also 
remarkable that the second proposition of the differences of the roots being equal to 
every number, odd or even, is immediately connected with Fermat’s second proposition 
of every number consisting of four squares or less. The connexion between the sum of 
the roots and Fermat’s first proposition was observed by me in the year 1854, and is 
mentioned in a paper which the Royal Society did me the honour to publish in the 
Philosophical Transactions for that year, vol. cxliv. p. 315, as Theorem C : see also p. 318. 
Before I proceed to show the other properties of The Square , I think it right to call 
attention to the manner in which a change in the roots of the four squares alters the 
sum of the squares themselves. This subject was touched upon by me in a paper pub- 
lished in the Philosophical Transactions for 1859, vol. cxlix. p. 49, in which it is stated, 
and a sort of proof (not a satisfactory one) given, that in some form of division into four 
squares the roots will be equal, will differ by 1, by 2, by 3, &c. 
I propose now to call attention to the effect, or result, of altering the roots so as to 
increase or decrease the sum of the squares. If the roots of two of the four squares 
that compose any odd number differ by n, and the larger of the two be increased by 1, 
and the smaller be decreased by 1, the sum of the squares will be increased by 2w+2. 
Let^> and p-\-n be the roots that differ by n , the sum of their squares will be 
2p 2 -f2 pn+n 2 . 
Ifp — 1 and {p-\-n )-\- 1 be squared, the sum of their squares will be 
p 1 — 2p + 1 -bp 2 + 2^m 4- n 2 + 2p -f- 2 n 1 , 
or 
2p 2 -}-2j0w-j-w 2 +2w-f-2 ; 
the increase is 2^+2. 
If n= 0 the increase is 2, 
if n— 1 the increase is 4, 
if n—1 the increase is 6, 
and so on, always twice the difference +2. 
There is a similar theorem for reducing the sum of the squares ; if p and p + n become 
