[QUOTE=LaurV;422988]You people have no patience. Now you see why the exponent has to be hidden :razz:[/QUOTE]

Who needs patience when you have the exponent :smile:

By the way, George was there any response from Numberphile?

[QUOTE=Dubslow;422972]The former.

Edit: Fun fact: the digit run 314159 occurs twenty two times in the decimal expansion, yet 3141592 doesn't occur. 2718281 occurs twice.[/QUOTE]

Hmm ... I get 3 occurrences of 2718281, starting at digits 5797998, 12817340 and 19391387. [Those counts treat the most-significant digit as the first and increase rightward, as in "Nth char in a file containing the decimal expansion," as displayed by many text editors in their positioning indicators.]

The decimal expansion also contains the exponents of the following known M-primes with p > 100000: 57885161 (once), 37156667 (once), 25964951 (once), 6972593 (3x), 3021377 (4x), 2976221 (3x), 1257787 (2x), 859433 (21x), 756839 (19x), 216091 (28x), 132049 (19x), 110503 (25x). Any p > 100000 not listed does not occur, including the as-yet-unannounced new one.

[QUOTE=ewmayer;422993]Hmm ... I get 3 occurrences of 2718281, starting at digits 5797998, 12817340 and 19391387. [Those counts treat the most-significant digit as the first and increase rightward, as in "Nth char in a file containing the decimal expansion," as displayed by many text editors in their positioning indicators.]

The decimal expansion also contains the exponents of the following known M-primes with p > 100000: 57885161 (once), 37156667 (once), 25964951 (once), 6972593 (3x), 3021377 (4x), 2976221 (3x), 1257787 (2x), 859433 (21x), 756839 (19x), 216091 (28x), 132049 (19x), 110503 (25x). Any p > 100000 not listed does not occur, including the as-yet-unannounced new one.[/QUOTE]Even more specific data. The query gets simpler.

I gave a quick shot from the hip at the [URL="http://maths-people.anu.edu.au/~brent/trinom.html"]P.Brent's irreducible trinomials[/URL] but the needed computation is enormous. Sieved just a little under s<10[SUP]5[/SUP]
(Recall that no irreducible trinomial was found for M[SUB]57885161[/SUB]. Tons of computrons were spent.)

[QUOTE=Dubslow;422972]Edit: Fun fact: the digit run 314159 occurs twenty two times in the decimal expansion, yet 3141592 doesn't occur. 2718281 occurs twice.[/QUOTE]
Even I know it now and I've never done any Mersenne crunching.

[QUOTE=Batalov;422996](Recall that no irreducible trinomial was found for M[SUB]57885161[/SUB]. Tons of computrons were spent.)[/QUOTE]

Does that mean that it is certain / proven that none exists?

[QUOTE=Dubslow;422972]The former.

Edit: Fun fact: the digit run 314159 occurs twenty two times in the decimal expansion, yet 3141592 doesn't occur. 2718281 occurs twice.[/QUOTE]


[QUOTE=ewmayer;422993]Hmm ... I get 3 occurrences of 2718281, starting at digits 5797998, 12817340 and 19391387. [Those counts treat the most-significant digit as the first and increase rightward, as in "Nth char in a file containing the decimal expansion," as displayed by many text editors in their positioning indicators.]

The decimal expansion also contains the exponents of the following known M-primes with p > 100000: 57885161 (once), 37156667 (once), 25964951 (once), 6972593 (3x), 3021377 (4x), 2976221 (3x), 1257787 (2x), 859433 (21x), 756839 (19x), 216091 (28x), 132049 (19x), 110503 (25x). Any p > 100000 not listed does not occur, including the as-yet-unannounced new one.[/QUOTE]
:doh!: :davieddy:

I was using the search function in my text editor, in which I had opened the "official" file hosted on mersenne.org, which has newlines every 100 chars... and the first instance of e that you listed starts two chars before a newline. My pi statistic may well be false too.

I can't even spill the beans correctly.

[QUOTE=ATH;423000]Does that mean that it is certain / proven that none exists?[/QUOTE]
Unless there was an error, yes, proven. They were rather dismayed.
"Uff-da! You've reached the end of the road."
They have an estimate of such outcome in their paper which is linked from Brent's page. It was a few percent iirc. But it happened. On average, they'd expect ~3.

[COLOR=Blue]EDIT: The certificates are provided, which are lists of factors; for example for r=57885161, and any s=1(mod 3), the line in the certificate reads, e.g.:
[B]10 2 p7[/B]
This (p7 is packed bits, coefficients {1,1,1} for {x^2,x,1}, and degree is 2) means that [B]x^2+x+1 | x^r+x^10+1[/B].
They have some enormous factors there, the largest being a [B]degree-7777044 poly[/B] | [B]x^[/B][/COLOR][COLOR=Blue][B][COLOR=Blue]57885161[/COLOR]+x^6341306+1[/B] (!!!).[/COLOR]

[QUOTE=Mark Rose;422960]My rule of thumb is that a super computer is on your desk in 10 years, your lap in 20, and your pocket in 30.[/QUOTE]

I'm not sure that's quite right; laptops and desktops are much closer together nowadays, with phones less far behind.

A single i7-5960X is 354 Linpack gigaflops, which is the score of the 500th-best supercomputer in November 2003. iPhone 6S+ linpack is 1.3 gigaflops, which would be 500th-best in November 1994 (and looks as if there's a software problem; 1.3 gigaflops is equivalent to the measured single-threaded Stream Triad bandwidth of 10.4GB/s, linpack ought to be able to go a lot faster by using the caches sensibly). So 10/12/20 seems a better sort of rule of thumb.

74207281
 

Verified, but no verifier.

So now we all know it of course. But why are there no verifications in the DB?

[url]http://www.mersenne.org/report_exponent/?exp_lo=74207281&full=1[/url]

[url]http://primes.utm.edu/primes/page.php?id=120909[/url]

Curtis Cooper again! Congrats to all involved. :smile: