2^R19-1
 

Just because R19 = 1111111111111111111 is prime,
is 2^R19 - 1 tested for primality (or (partially) factored)?

(I tried factoring 2^R4-1 and the applet was working on a C278
for 3 minutes when I left it. And 2^R19-1 is way bigger ...)

[quote=davar55;135269]Just because R19 = 1111111111111111111 is prime,
is 2^R19 - 1 tested for primality (or (partially) factored)?

(I tried factoring 2^R4-1 and the applet was working on a C278
for 3 minutes when I left it. And 2^R19-1 is way bigger ...)[/quote]
According to [URL="http://v5www.mersenne.org/report_factors/?exp_lo=524287&exp_hi=524287&exp_date=&dispdate=1&B1=Get+Factors"]this page on PrimeNet v5[/URL], 2^524287-1 has four factors; ECM progress can be viewed at the [url=http://v5www.mersenne.org/report_ECM/]PrimeNet v5 ECM report page[/url] (right now it's the last exponent listed on that page). :smile:

[QUOTE=davar55;135269]Just because R19 = 1111111111111111111 is prime,
is 2^R19 - 1 tested for primality (or (partially) factored)?

(I tried factoring 2^R4-1 and the applet was working on a C278
for 3 minutes when I left it. And 2^R19-1 is way bigger ...)[/QUOTE]

Will Edgington's lowm.txt file also shows a C278 remaining for 2^R4-1. The R19 is beyond the range of lowm.txt, but Will keeps factors for all Mersenne numbers. You could email him to find out about any known factors. If you find any factors, you should report them to him.

[url]http://www.garlic.com/~wedgingt/mersenne.html[/url]

Factordb now reports the following complete factorization for 2^R4-1 :

2^1111-1<335> = 23 · 89 ·
7432339208719<13> ·
341117531003194129<18> ·
457892136759270759480793<24> ·
9401699217...57<130> ·
1245111485...51<148>

The last two factors comprise the C278.

[QUOTE=mdettweiler;135281]According to [URL="http://v5www.mersenne.org/report_factors/?exp_lo=524287&exp_hi=524287&exp_date=&dispdate=1&B1=Get+Factors"]this page on PrimeNet v5[/URL], 2^524287-1 has four factors;[/QUOTE]

When I counted the number of factors on that page, there were five of them, not four.

So what's the big deal with 2^R19-1 ?
Granted, 2^1111111111111111111-1 doesn't have any known factors.
But so does 2^1111111111111111007-1, and 2^1111111111111111171-1, and many others of the same size. Try them.

On the other hand,
26666666666666664889 | M1111111111111111037
3607706666666666426032633 | M1111111111111111037

Take a prime p large enough and you will frequently be unable to find factors of Mp, but only because prime density for 2kp+1 will be low and exponentiations are slowish (especially with your famous pocket calculator). So what?

No factor is known for M{11!+1} and 11!+1 << R19. And the same for ~30% of even smaller primes! Where is the element of surprise or was it "just because"?

No need for negativity, in this search we're on the same team.

R19 is the smallest odd-indexed repunit which is prime, hence
2^R19 - 1 is similar in form to the Mersenne Primes and hence
interesting, at least to me.

[QUOTE=davar55;394340]No need for negativity, in this search we're on the same team.

R19 is the smallest odd-indexed repunit which is prime, hence
2^R19 - 1 is similar in form to the Mersenne Primes and hence
interesting, at least to me.[/QUOTE]I'm almost tempted to go looking for factors. Unfortunately, I'm also interested in seeking factors of other integers and my resources are limited.

[QUOTE=xilman;394527]I'm almost tempted to go looking for factors. Unfortunately, I'm also interested in seeking factors of other integers and my resources are limited.[/QUOTE]

It is an isolated curiosity/choice and is so far beyond the capabilities of existing methods that I see no purpose
in such a pursuit. Even if one succeeds in finding a small factor, the obvious question is: "so what?"

[QUOTE=R.D. Silverman;394534]It is an isolated curiosity/choice and is so far beyond the capabilities of existing methods that I see no purpose in such a pursuit. Even if one succeeds in finding a small factor, the obvious question is: "so what?"[/QUOTE]Exactly.

The only reason I'd do it is out of idle curiosity, in much the same way that I'd turn over one rock out of many to see what might be underneath it.

[QUOTE=xilman;394548]Exactly.

The only reason I'd do it is out of idle curiosity, in much the same way that I'd turn over one rock out of many to see what might be underneath it.[/QUOTE]

[CODE]M1111111111111111111 has 0 factors in [2^1, 2^90-1].
M1111111111111111111 has 0 factors in [2^90, 2^92-1].
M1111111111111111111 has 0 factors in [2^92, 2^100-1].[/CODE]
You can avoid these particular rocks