[seqfan] Converse
Tomasz Ordowski
tomaszordowski at gmail.com
Tue Apr 17 13:36:28 CEST 2018
Hello!
Let us define:
Numbers n such that 2^n == 2 (mod n).
Theorem: if n is such a number,
then 2^n-1 is such a number.
Problem: is the reverse theorem true?
The converse holds up to n = 10000.
Generally:
Let B be the set of natural numbers n such that
(b^n-1)/(b-1) == 1 (mod n)
with a fixed integer b > 1.
Conjecture:
The number (b^k-1)/(b-1) belongs to B if and only if k belongs to B.
Note: This can be extended to bases b < -1.
Do you have an idea how to prove my conjecture?
Professor Pomerance does not know a proof.
Please reply, it's important to me!
Sincerely,
Thomas Ordowski
________________________
Steuerwald's theorem (1948):
If k is a psp(b) and gcd(k,b-1)=1, then (b^k-1)/(b-1) is a psp(b).
Here a composite k is a psp(b) if and only if b^k == b (mod k).
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