[seqfan] Sequences in relation to Crocker's theorem

[Tomasz Ordowski]

Dear readers!

   Roger Crocker (1971) proved that,
if n > 2 and a <> b, then 2^(2^n)-1 - 2^a - 2^b is not prime.

   If n > 2, then the largest prime Q(n) (or the smallest R(n))*
of the form 2^(2^n)-2^m-1 is a de Polignac number. For n > 0,
A232565(n) is the smallest k such that 2^(2^n)-2^k-1 is prime.
Conjecture: if n > 6, then the prime Q(n) is a dual Riesel number:
both |Q(n)-2^k| and Q(n)2^k-1 are not prime for every natural k.
For example, the (largest) prime Q(7) = 2^(2^7) - (2^18+1)
may be a dual Riesel number. Note that  A232565(7) = 18 and
2^18+1 is the smallest de Polignac number of the form 2^k+1.

Let a(n) be the smallest k > 0 such that 2^(2^n)+2^k-1 is prime.
Data (for n >= 0): 1, 1, 1, 1, 1, 4, 21, 115, 96, 96, 432, 271, 712, ...
These (Fermat) primes P(n) = 2^(2^n)+1 for n < 5; and next
primes; P(5) = 2^(2^5)+2^4-1,  P(6) = 2^(2^6)+2^21-1, ...
Then let b(n) be the smallest m such that P(n)+2^m is prime.
Data (for n >= 0): 1, 1, 1, 9, 1, ?, ?? Maybe someone will find more;
b(5) > 6666, if it exists. If not, then P(5) is a dual Sierpinski number,
by the dual Sierpinski conjecture: P(5)2^k+1 are composite. Right?
Conjecture: if n > 5, then the prime P(n) is a dual Sierpinski number,
i.e. both P(n)+2^k and P(n)2^k+1 are composite for every natural k.
For example, the prime P(6) is the next candidate. Try it.

Note: it is not known whether these dual Sierpinski numbers
and these dual Riesel numbers have any covering sets.
The prime divisors of 2^(2^n)-1 are important here!

See https://oeis.org/A006285 (de Polignac numbers).
See https://oeis.org/A101036 (Riesel numbers with covering).
See https://oeis.org/A076336 (provable Sierpinski numbers).
Cf. https://oeis.org/A156695 and https://oeis.org/A232565
Look at https://oeis.org/history/view?seq=A369375&v=22
and at https://oeis.org/history/view?seq=A369378&v=39
also https://oeis.org/history/view?seq=A232565&v=33
(these last three are my recent drafts).

Best regards,

Thomas Ordowski
___________________
(*) The smallest number R(n) = 2^(2^n-1)-1 for n > 2
(by Crocker's theorem) is a de Polignac number.
Conjecture: if n > 6, then R(n) is a Riesel number.
For example, the double Mersenne prime 2^(2^7-1)-1
(by the dual Riesel conjecture) may be a Riesel number,
i.e. both |R(7)-2^k| and R(7)2^k-1 are not prime for all k > 0.

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