000RM.dvi

18.3 Sam Yates’ repunit riddles 521

8.Find a pair of repunits whose product is a 100-digit palindrome.^7

9.If a Mersenne numberMp=2p− 1 is prime, is the corresponding

repunitRpalso prime?^8

10.What is the smallest repunit divisible by the square of 11? What is

the smallest repunit divisible by the square ofR 11?

In general, the smallest repunit divisible by the square ofRnisRN,

whereN=nRn.

(^7) IfRp·Rqhas 100 digits,p+q=101. Supposep≤qandp=9k+mfor 1 ≤m≤ 9. Since the
product is a palindrome, it cannot containAandB. We must havek=0andp≤ 9. For anyp=2, ...,9,
the productRp·Rqis the palindromep 102 − 2 p 1 , where
p=
(
(^12) k ifp=2,
12 ···(p−1)(p−1)32 if 3≤p≤ 9 ..
(^8) M 3 =2 (^3) −1=7is prime butR 3 = 111 = 3× 37.