[seqfan] The Repunit Theorem

[Tomasz Ordowski]

Dear SeqFan,

I present a new formulation of the repunit theorem:

Let integer b <> 1 and n be a positive integer.

Define R_b(n) = (b^n-1)/(b-1) = N.

(*) Then R_b(N-1) == 0 (mod N) if and only if N == 1 (mod n).

(**) Then R_b(N) == 1 (mod N) if and only if N == 1 (mod n).

Now the proof is obvious.

It is an inexhaustible source of new sequences and comments.

For example, http://oeis.org/A015919, a new comment:
N = 2^n-1 is a term if and only if n is a term.
Cf. Max Alekseyev's comment.

Best regards,

Thomas

P.S. I am waiting for other ideas.




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