14.2 The repunits 417
14.2 The repunits ..................
The repunitRnis the number whose decimal representation consists of
a string ofnones. Thus,Rn=1n.
Notations
A:=123456790,
B:=098765432,
(^1) k:=1k− 1 ,
(^2) k:=12k,
mk:=12·(m−1)mk(m−1)··· 2 , for 3≤m≤ 9.
Theorem 14.1.Letp=9k+m,k≥ 0 , 1 ≤m≤ 9 .Forq≥p,
RpRq=Akmq−p+1Bk 1.
In particular,
(Rp)^2 =AkmkBk 1.