BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 93 
Hence m*=97r+1 or 97; the former when m is prime to 9, the latter when it has 
a common divisor with it. This is a form not given by Bartow. 
Prop. XII. 
In the series of m* powers of successive integers, beginning from 1, the ulti- 
mate sums recur after »—1 terms. 
If m be odd, the ultimate sums of any two terms, whose roots together equal 
n—I, are either together equal n—1, or are each n—1. 
If m be even, the ultimate sums of such complemental terms are identical. 
After what has been proved, the demonstration of these is so easy that it is 
unnecessary to give it. 
Ex. Series of 2d powers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121. 
Ultimate sums ES lh Ol A Qu id. 
Series of 5th powers 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 9°, 10°. 
Ultimate sums eam on ei Dir pls 2. 9, 4, See OH tT 
Here, in the ultimate sums of the squares, we have Ist and 8th, 2d and 7th, &c. 
identical. In the ultimate sums of 5th powers, the Ist + 8th = 9, 2d + 7th = 9, 
and so on. 
It is worthy of notice, though rather out of place, that if, in the series of 
5th powers, instead of taking the sums, we take the difference between the sums 
of the odd and even digits, the difference will in every case be 1. This property 
is proved generally by Bartow, in his Theory of Numbers, p. 172, in this form 
m—1 
that 2 7 , where misa prime number, is of the form am=+ 1. 
Ex. To illustrate this, and the property of sixth powers mentioned in the 
XIth Prop., we shall take the 5th and 6th powers of 5 and 8. 
5'— 3125 therefore d,=(5+1)— (34 2)—1. 
5°= 15625 therefore S,=19, S,=10, S,=1. 
8= 32768 therefore d,=(8+7+3)—(6+2)=10, d,=1. 
8°= 262144 therefore s,=19, S,=10, S,=1* 
Cor. From the property above demonstrated of the sixth powers of numbers 
prime to 9, it follows, that for every such base the seventh power has its ultimate 
sum equal to the base ; that is, that e7=m.9+a. For a=p.9+1,.. a’ =pa9-+a. 
Ex. 5'= 78125, S,=23, S,=5. 
87 = 2097152, Ss, = 265) 1S, = 8. 
* In these equations, d,, d,, &c., express the Ist, 2d, &c., differences between the sums of the 
odd and even digits; S,, S,, &c., express the Ist, 2d, &c., sums of all the digits. 
