Provable Riesel numbers (A076337).

[David Wilson]

It looks like A076337 has been waiting a long time for an extension.
I submit the following list of provable Riesel numbers.  It is rather
longer than it needs to be for the OEIS, but I am rather ill right now,
and I fell asleep with the program running.

A Riesel number is number n such that (2^k)n-1 (k >= 1) is always
composite.  n is a provable Riesel number if a periodic sequence
of prime divisors p with p(k) | (2^k)n-1 can be found.  If no such
sequence exists, a probabilistic argument indicates that some
(2^k)n-1 should be prime, but there seems to be no way to settle
the question except to find the prime.

A similar and older problem, involves the provable Sierpinski
numbers of the form (2^k)n+1.  At www.seventeenorbust.com,
there is a concerted computational effort to eliminate the 17
Sierpinski candidates less than the smallest proved Sierpinski
number 78557.  10 have been eliminated (the 10th is described
at mathpuzzle.com, Mathworld apparently hasn't caught up).

The conjecture is that all Riesel numbers are in fact provable.
The smallest known provable Riesel number is 509203, there exist
many smaller undecided Riesel candidates.  The presumption is
that these candidates are not Riesel.
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Here is my list of proved Riesel numbers.

509203 762701 777149 790841 992077 1106681 1247173 1254341 1330207
1330319 1715053 1730653 1730681 1744117 1830187 1976473 2136283 2251349
2313487 2344211 2554843 2924861 3079469 3177553 3292241 3419789 3423373
3580901 3661529 3661543 3781541 3784439 4384979 4442323 4485343 4506097
4507889 4570619 4626967 4643293 4953397 5049251 5050147 6055001 6610811
6975809 7106977 7117807 7576559 7629217 7790113 8010517 8086751 8101087
8252819 8253043 8482363 8643209 9053711 9053767 9203917 9375479 9545351
9560713 9666029 10157893 10219379 10280827 10581097 10609769 10645867
10702091 10913233 10913681 11124703 11694013 11942443 11947511 12000697
12176887 12431983 12439151 12515017 12515129 12915463 12915491 12973451
13006807 13161283

Other references on the Web:

http://mersenneforum.org/showthread.php?t=2180 agrees with
my first 5 values.  I suspect the poster is in possession of
additional values.

http://www.15k.org/lowweight.htm agrees with my first 10 values.

What kind of work is going on at these sites?  What is "15k"
that I hear mentioned in these posts?

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I would like to ask someone else to verify these numbers before I
submit anything to the OEIS.  Upon verification, I will submit
A076337.  I will also ask that NJAS hold or publish the full list
above so that it does not get lost.  Mathworld might also want to
update its article on Riesel numbers once verification is done.

In my own testing, I used a list of 181 primes <= 10^5 with ord(2,p)
5-smooth.  If it helps in verification, the only primes I actually
needed were

3 5 7 11 13 17 19 31 37 41 73 97 109 151 241 257 433 601 1801

For these primes, ord(2,p) divides 144.

As with the Sierpinski numbers, I cannot absolutely vouch for the
completeness of the list, since there may be larger primes or periods
involved than I tested for.