000RM.dvi

418 Numbers with many repeating digits

14.3 Squares of repdigits ..................

In the decimal representations of integers, we writeanfor a string ofn

digits each equal toa.

Theorem 14.2.Forn≥ 2 ,

(3n)^2 =1n− 108 n− 19 ,

(6n)^2 =4n− 135 n− 16 ,

(9n)^2 =9n− 189 n− 11.

Proof.The last one is easiest.

(9n)^2 =(10n−1)^2

=10^2 n− 2 · 10 n+1

=10n(10n−2) + 1

=9n− 180 n− 11.

From this we obtain the square of (^3) nby division by 9, then the square of
(^6) nby multiplication by 4.
Theorem 14.3. Letn =9k+m,k ≥ 0 , 1 ≤ m ≤ 9 .Fora =
1 , 2 , 4 , 5 , 7 , 8 ,
(aRn)^2 =AkmBkc,
whereA,Bandcare given by
a A B c
1 123456790 098765432 1
2 493827160 395061728 4
4 197530864 580246913 5
5 308641975 469135802 6
7 604938271 839506172 9
8 790123456 320987654 4
andmis given by
a 1 2 3 4 5 6 7 8 9
21 4812 49281232 493728123432 1234543249381728 12345654324938261728 493827061728123456765432 1234567876543249382715061728 12345678987654324938271595061728
54 21 302193 3080219713 19749133085802 308635802197526913 3086413580219753046913 19753082469133086419135802 197530860246913308641969135802 1975308638024691330864197469135802
87 64 774592 7885460372 60481727899654 790107654604926172 7901218765460493706172 60493815061727901232987654 604938259506172790123440987654 6049382703950617279012345520987654