52 SIB r. POLLOCK ON THE BELATION BETWEEN BOOTS OP 
1 2 3 5 -here are no equal roots ; but the other mode has the roots 1 , 1 , 1 . 6 , wMch 
1, 2, 3, 6 , here are n q , 1 ,^ fom 
furnish two equal roots. In i r. u i, pasv to nroTe that eveiw odd 
n2 4-6^4-2c^ that is, has 2 of its roots equal*; also it is eas^v to p 
numhet wiU have 2 of its roots differing hy 1 , that is, will he of the form 
u f fVip form 4^^+l may be composed of 3 squares (4 are not nece^ 
"aldToIly one of them can be an odd square (for if there were 3 odd squares r.s 
forai must be 4rt+ 3), i 
te+l=4a’+45=+4<f+dc+l, 
and n (any number) =:a"+5"+c +c, 
and 2 %+l (any odd number) ^ 
=2a^+25^+2c^+2c+l = («+^)'+(«“^)'+'^+<^"+^^‘’ 
if 2 of the roots must he equal, and ^ ^ 
thus ‘25 denotes 25 with roots 0, 0, 3, 4 or -2, 1, 4, 2 ; '26 may be -o, 0 , 0, 4, (.» +1) 
“wto’to complete the proof, I must now call attention more particularly to the 
nmuerties of the two series already mentioned. i o c 
' The fimt is 1. 3, 9, 19, &c., general term (2»»+l) ; it mcreases by the numbers .. 6 . 
“xre’strrnri^-, general term ( 2 .+ 2 »+l)r the rrrcrease is by the 
-r t \t\trt"of" — whose roots may differ by the crcn 
"TteTecond is the series of the least odd numbers whose 
numbers. This is quite obvious: the number »(2»^+l) 
number with the same difference of roots must be g'-<=ater, 1 ). - - + ' 
so ( 2 m'+ 2 «+ l), the difference of whose roots may be ( 2 »+l). . + 
least number that can have that difference of roots. ,m, nauer. They 
These two series are mentioned as “i/iciwo senes uiing „ated f' 044 
may be formed together by increasing from 1 , by 
6 , 6 , &c.) : thus, 1, 3, 5, 9, 13, 19, &c. Thrs sen^s - f ^ ‘ ^ p Pave 
but 1 belongs to both— as •! to the one, as 1 to the ot ei. Put ^ ^ 
suggested above and the roots below, the fost senes will appear thus. 
01 ^3 ‘9 °19 
0 , 0 , 1,0 - 1 , 0 , 1,1 - 2 , 0 , 1, 2 - 0 , 0 , 1,0 
* Legesdue, Theorie des Nombres, 1st edit. p. 186 ; 2nd edit. p. 391. 
