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Old 2020-07-22, 23:28   #1
JeppeSN
 
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"Jeppe"
Jan 2016
Denmark

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Question Minimal Riesel number with (apparently) no covering

Hello all,

For the purpose of this post, I define a Riesel number as a positive odd number k for which k*2^n - 1 is composite for all n > 0. Note that I do not require the existence of a (full) covering set here, and this is the whole point.

Explicit, proven Riesel numbers for which no full covering set is known (and for which we conjecture that no finite, full covering exists), can be given. For example:

k = 1469583304447640330447613742^3

This example was found by user Gelly (who had other contributions in the thread Numbers Sierpinski to multiple bases). For details, you can see my novel post on Stack Exchange if you like.

I want smaller examples!

My question here is: What is the smallest Riesel number without a covering set? Equivalently the smallest Riesel number k for which the least prime factor of k*2^n - 1 is unbounded as n goes to infinity.

I want a number k that is proven Riesel with a combination of algebraic factorizations and a partial covering set, and for which it is not "easy" to augment the partial covering to a full covering (so that the algebraic factorization becomes unnecessary). I know we cannot prove that no full covering exists, so that will just be a plausible conjecture.

/JeppeSN

PS! For the analogous question for Sierpiński numbers, I am quite sure the answer is k=44745755^4. That is, it is a conjecture (at least to me) that 44745755^4 is the smallest Sierpiński without a covering. I want the conjectural smallest Riesel.
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