10^(n^2) has n^2 + 1 digits.
So we want n*m - m = n^2 + 1.
m*(n-1) = n^2 + 1
m = (n^2 + 1) / (n - 1)
Pairs (m,n):
(5,2) and (5,3)
RPN to find m from n:
==*1+w1-/
i.e.,
duplicate the entry twice and multiply bottom two
add 1
swap with third copy
subtract 1
divide
produces m from n
The RPN was used in a calculator program to produce a table.
N M
1 err
2 5
3 5
4 5.666666666666667
5 6.5
6 7.4
7 8.333333333333334
8 9.285714285714287
9 10.25
10 11.22222222222222
11 12.2
12 13.18181818181818
13 14.16666666666667
14 15.15384615384615
15 16.14285714285714
16 17.13333333333333
17 18.125
18 19.11764705882353
19 20.11111111111111
20 21.10526315789474
21 22.1
22 23.0952380952381
23 24.0909090909091
24 25.08695652173913
25 26.08333333333333
26 27.08
27 28.07692307692308
28 29.07407407407407
29 30.07142857142857
30 31.06896551724138
31 32.06666666666667
32 33.06451612903226
33 34.0625
34 35.06060606060606
35 36.05882352941177
36 37.05714285714286
37 38.05555555555556
38 39.05405405405406
39 40.05263157894737
40 41.05128205128205
...
1381 1382.001449275362
1382 1383.001448225923
1383 1384.001447178003
1384 1385.001446131598
1385 1386.001445086705
1386 1387.001444043321
1387 1388.001443001443
...
Edited on December 23, 2015, 11:06 am
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Posted by Charlie
on 2015-12-23 11:01:08 |
solution
