BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 93 



Hence m«=9 r+ 1 or 9 r ; 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 Barlow. 



Peop. 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 

 IT-i, are either together equal n-1, or are each »-l. 



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 1, 4, 9, 7, 7, 9, 4, 1, 9, 1, 4. 



Series of 5th powers 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 9^, 10'. 

 Ultimate sums 1, 5, 9, 7, 2, 9, 4, 8, 9, 1. 



Here, in the ultimate sums of the squares, we have 1st and 8th, 2d and 7th, &c. 

 identical. In the ultimate sums of 5th powers, the 1st + 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 suras, 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 Barlow, in his Theory of Numbers, p. 172, in this form 



m — 1 



that X ^ , where m is a prime number, is of the form am±l. 



Ex. To illustrate this, and the property of sixth powers mentioned in the 

 Xlth Prop., we shall take the 5th and 6th powers of 5 and 8. 

 55 = 3125 therefore rf^ = (5 + 1) - (3 + 2) = 1. 

 5«= 15625 therefore 5i= 19, .S'„ = 10, ^3=1. 

 8' = 32768 therefore rf^ := (8 + 7 + 3) - (6 + 2) = 10, rf„ = 1. 

 8" =262144 therefore S^ = 19, S„=10, S^=l * 

 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 a' = j».9 + a. For a" =p.9 + 1, .-. a'=pa.9 + a. 

 Ex. 5^= 78125, S^=23, S„ = 5. 



8^=2097152, .S'i=26, S^ = 8. 



ft 



* In these equations, d^, d„, &c., express the 1st, 2d, &c., differences between the sums of the 

 odd and even digits; S^, S^, &c., express the 1st, 2d, &c., sums of aU the digits. 



