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 Repunit Rigor II (Posted on 2013-10-01)
sod(n) denotes the sum of the digits of a base ten positive integer n, and:
Rt = 11...11 (the digit 1 repeated precisely t times.)

Determine the values of t for which:
`sod(Rt2) = (sod(Rt))2`

 No Solution Yet Submitted by K Sengupta Rating: 5.0000 (1 votes)

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 computer exploration | Comment 1 of 3

10   for T=1 to 46
20     Rt=Rt*10+1
30     Rtsq=Rt*Rt
40     Rtsqs=cutspc(str(Rtsq))
45     Tot=0
50     for I=1 to len(Rtsqs)
60       Tot=Tot+val(mid(Rtsqs,I,1))
70     next
80     Tot2=0
90     Tsq=T*T
125     print T,Tot;Tsq;:if Tot=Tsq then print " *":else print
130   next

finds

t      LHS RHS
1       1  1  *
2       4  4  *
3       9  9  *
4       16  16  *
5       25  25  *
6       36  36  *
7       49  49  *
8       64  64  *
9       81  81  *
10      82  100
11      85  121
12      90  144
13      97  169
14      106  196
15      117  225
16      130  256
17      145  289
18      162  324
19      163  361
20      166  400
21      171  441
22      178  484
23      187  529
24      198  576
25      211  625
26      226  676
27      243  729
28      244  784
29      247  841
30      252  900
31      259  961
32      268  1024
33      279  1089
34      292  1156
35      307  1225
36      324  1296
37      325  1369
38      328  1444
39      333  1521
40      340  1600
41      349  1681
42      360  1764
43      373  1849
44      388  1936
45      405  2025
46      406  2116

The solutions with LHS=RHS are marked with * (t=1 thru 9).

After t=9, the RHS starts to exceed the LHS at a growing rate. By t=500, the LHS = 4480 and the RHS = 250000.

 Posted by Charlie on 2013-10-01 19:35:16

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