BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 91 



To iUustrate this and some of the succeeding propositions, I shall here 

 introduce a table of successive powers of digits prime to 9, with their ultmiate 



sums. 



Base 2 



Base 4 



I Powers 2, 4, 8, 16, 32, 64, 128, &c. | ^^^^ ^^^^^ ^^^^^ g ^^^^^ 

 \ Sums 2, 4, 8, 7, 5, 1, 2, &c. J 



f Powers 4, 16, 64, 256, 1024, &c. | ^^^^ ^^^^^ ^^^^^ 3 ^^^^^^ 

 1 Sums 4, 7, 1, 4, 7, &c. J 



f Powers 5, 25, 125, 625, 3125, 15625, 78125, &c. ] Sums recur after 



^^^ ^ ftnnis 5, 7, 8, 4, 2, 1, 5, &c. j 



\Sums 5, 7, 8, 4, 2, 1, 



6 terms. 



Base 



I Powers 7, 49, 343, 2401, 16807, &c. 1 ^^^^ ^^^^^ ^^^^^ 3 ^^^^^ 

 [Sums 7, 4, 1, 7, 4, &c. j 



Base 8 



f Powers 8, 64, 512, 4096, &c. 1 ^^^^ ^^^^ ^^^^^ ^ terms. 

 \Sums 8, 1, 8, 1, &c. J 



■ In this table, we may observe that in every case the sum of the digits recurs, 

 but at different intervals. Next, if we take two complementary bases, as 5 and 4, 

 we find in the lines expressing the sums, that the first terms are respectively 

 4 and 5, the 2d terms 7 and 7, the 3d 1 and 8, and so on; as was proved gene- 

 raUy in the last proposition. Lastly, we may observe -that the digits 3, 6, 9, that 

 is »-l, and the digits having a common divisor with n-l, never occur among the 

 sums. It remains, then, for us to point out the reason of this last mentioned 

 fact, and to discover the principle which fixes the period of recurrence. 



Pkop. X. 



Every power of a number prime to ^m, must have the sum of its digits also 



prime to w— 1. j? i. 1 + 



Let m, which is prime to ^^, be reduced to its prime factors, or let 



m^a.b.c, &C. 



Then »/ = (a'^.b"./ &c.) Here m" has no possible divisors except a, b^ &c., 

 and by hyp. none of these are divisors of i^^, therefore m' is prime to n-1. 



Now let,/ =q:^l + r. Here q^^ contains all the divisors of n-1. If, 

 therefore, r contains any of those divisors g^^l + r, or m" contains such divi- 



VOL. XVI. PABT II. 



