Gelfand's Question

	Sloane's
1	A000027	2	3	4	5	6	7	8	9
2	A002993	4	9	1	2	3	4	6	8
3	A002994	8	2	6	1	2	3	5	7
4	A097408	1	8	2	6	1	2	4	6
5	A097409	3	2	1	3	7	1	3	5
6	A097410	6	7	4	1	4	1	2	5
7	A097411	1	2	1	7	2	8	2	4
8	A097412	2	6	6	3	1	5	1	4
9	A097413	5	1	2	1	1	4	1	3
10	A097414	1	5	1	9	6	2	1	3

Consider the leftmost (i.e., most significant) decimal digit of the numbers , , ..., . Then what are the patterns of digits occurring in the table for , 2, ... (King 1994)? For example,

1. Will the digit 9 ever occur in the  column? The answer is "yes," in particular at values , 63, 73, 83, 93, 156, 166, 176, ... (OEIS A097415. This problem appears in Avez (1966, p. 37), where it is attributed to Gelfand.

2. Will the row "23456789" ever appear for ? None does for . If so, will it have a frequency? If so, will the frequency be rational or irrational?

3. Will a row of all the same digit occur? No such example occurs for .

4. Will the decimal expansion of an 8-digit prime ever occur? (The answer is "yes," in particular at values , 11, 21, 44, 55, 81, 90, 118, 126, ... (OEIS A097616), corresponding to the primes 23456789, 21443183, 21442591, 19351159, ... (OEIS A097617).

Amazingly, this problem is isomorphic to Poncelet's porism (King 1994).

REFERENCES:

Avez, A. Ergodic Theory of Dynamical Systems, Vol. 1. Minneapolis, MX: University of Minnesota Institute of Technology, 1966.

King, J. L. "Three Problems in Search of a Measure." Amer. Math. Monthly 101, 609-628, 1994.

Previato, E. "Featured Review: CRC Concise Encyclopedia of Mathematics. Second Edition." SIAM Rev. 46, 349-354, 2004.

Sloane, N. J. A. Sequences A000027/M0472, A002993/M3348, A002994/M4468, A097408, A097409, A097410, A097411, A097412, A097413, A097414, A097415, A097616, and A097617 in "The On-Line Encyclopedia of Integer Sequences."