SQUAEES INTO WHICH ODD NUMBEES MAT BE DIVIDED. 57 
where obviously the remainders equidistant from the middle term are equal to one 
another. 
Now the difference between the adjoining terms of either of the 2 series will always 
be divisible by 2, the difference between the alternate terms will be divisible by 4, the 
difference between 2 terms that are m apart will be divisible by 2m. 
For the wth term of the series 1, 3, 9, 19, &c. is 27v‘-\-\, and the {n-^myih. term is 
and their difference, 4wm+2m®, is divisible by 2m. So in the other 
series (1, 5, 13, 25, &c.) the wth term is 2w®+2w+l, and the (w+m)th term is 
{n-\-my-\-{n-{-m-{-Vf"2n^-\-2n-\-l-^2m?-\-^mn-\-2m-, 
and their difference, 2m^-\-^mn-^2m, is also divisible by m. An arithmetical series 
may therefore always be formed, which will give the required difference of the roots 
according to the distance of the pair of terms from the middle term. 
The general result therefore is, that if any 2 odd numbers be assumed, they will 
either have this relation to each other of the roots of the 4 square numbers into which 
they may be divided, or a third odd number may be found, which will connect them 
together by having that relation to each. 
If the 2 odd numbers be the result of an addition of the same even number to any 
2 terms of either of the two series, they will have this relation of the roots, and the dif- 
ference of the exterior roots will depend upon the distance of the 2 terms from each 
other ; and conversely, if any 2 odd numbers have this relation of the roots, they are 
derived from 2 terms of the same series by the addition of the same even number to 
both ; but if the 2 odd numbers have not this relation of their roots to each other, then 
a third odd number may be found having that relation to each of them. 
Assume any 2 odd numbers as 13 and 105, deduct 12 from each of them so as to 
reduce the smaller to 1, and the other to 93, the next term in either series less than 93 
is 85, 85+8=93. Select that term in the series to which 85 belongs, which by the 
addition of 8 becomes a term in the other series, this is 25, 25 + 8 = 33, and 33 is the 
number which connects 1 with 93 ; for 
and 803 
0,0, 1,0 -4, 0,1, 4 
have the relation which appears from their roots, and 
^33 and 1393 
-3, 2, 2, 4 - 6 , 2, 2, 7 
-2, 0,2, 5 -5, 0,2, 8 
have a similar relation in two ways; then add 12 to 1, to 33, and to OS’; and 13, 45 and 
105 will have this relation among their roots; 
°13 
+ 2 , 1 , 2 , 2 
0, 2, 3, 0 
«45 
-2, 1,2, 6 
-4,2, 3,4 
’45 
-3, 2, 4, 4 
-2,0, 4,5 
- 1 , 2 , 2, 6 
^no5 
- 6 , 2, 4,7 
-5, 0, 4,8 
-4, 2, 2, 9 
MDCCCLIX. 
