[seqfan] Related to Wolstenholme's theorem
Tomasz Ordowski
tomaszordowski at gmail.com
Wed Jun 24 13:14:22 CEST 2020
Dear readers!
Let F(n) = Sum_{k=1..n} 2^{k-1}/k = N(n)/D(n):
1/1, 2/1, 10/3, 16/3, 128/15, 208/15, 2416/105, ...
Theorem: If p > 3 is prime, then p F(p) == 1 (mod p^3);
equivalently p N(p) == D(p) (mod p^4), since p | D(p).
Conjecture: For n > 3, if n N(n) == D(n) (mod n^4),
then n is prime. If so, there are no such pseudoprimes.
Are there the weak pseudoprimes, except n = 49 ?
Composites n such that n N(n) == D(n) (mod n^3):
7^2, 16843^2, 2124679^2, ... To be confirmed.
Primes p such that p N(p) == D(p) (mod p^5)
are 7, 16843, ? Cf. A088164:
https://oeis.org/A088164
Is p = 2124679 as well?
Best regards,
Thomas Ordowski
_______________________
https://en.wikipedia.org/wiki/Wolstenholme%27s_theorem
Note that N(n) = A108866(n)/2 : https://oeis.org/A108866
and D(2n+1) = D(2n+2) = A229726(n) : https://oeis.org/A229726
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