PROCEEDINGS 



OF THE 



ROYAL SOCIETY OF EDINBURGH. 



VOL. II. 1845-6. No. 27. 



Sixtt-Third Session. 



First Ordinary/ Meeting, 1st December 1845. 



Sir T. M. BRISBANE, Bart., President, in the Chair. 



The following Communications were read : — 



1. On the Sums of the Digits of Numbers. By the Right 

 Rev. Bishop Terrot. 



It was shewn, that, if the ultimate sums of the digits of the terms 

 of any arithmetic series, whose difference is prime to the local value 

 of the notation employed, minus 1, be taken, such sums will range, 

 without any recurrence, through all the digits of the notation, and 

 then recur in the same order as before. It was then shewn, that 

 in any integer series formed upon a given law, the sums of the digits 

 of the terms will have a fixed period of recurrence. This was proved 

 in polygonal and figurate numbers, in the series of squares, cubes, 

 &c., in the successive powers of a given root; in the series whose 

 general term is m.m + 1 . . . m + r — 1; and in that whose gene- 

 ral term is a; "»+ a a; ""-^ . ... +1. 



It was shewn, that whenever the variable or number of the term 

 occurs as a multiplier, the sums recur after n — 1 terms, with cer- 

 tain definite relations between the intermediate terms. When the 

 variable occurs as an index, then the recurrence was shewn to take 

 place at shorter intei'vals ; in the decimal notation, according as the 

 root was of the form 3 />, or 3 p =t: 1. From the determinatiun of 

 the diflbrent periods of recurrence in the last case, it was shewn 

 that every sixth power is of the form 9 n, or 9 71+ 1 ; and that 



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