Generalized Repunit primes
 

Has anyone tried to create a distributed computing project for Generalized Repunit Primes? Andy Steward had a webpage on this subject that seems to be discontinued. I miss it. It seem like a project that would be easy to understand and share:

So ill state it again here:
Repunits are 'repetitions of the unit', a series of ones. Eleven is a prime and written R2. One hundred eleven is 3X37, a composite written R3. 1111 is R4 and is composite. The prime repunits are quite rare: R2 ,R19, R23, R317, R1031 have been discovered so far--and that's it! Two other Rs are pending, suspected but not yet shown to be prime. It is necessary but NOT sufficient for the R number to be prime for the Repunit to be prime .What a suprise to think the simple number ONE holds any mysteries.
In bases other than 10 the Reps are also rare. In base two...well, you know about that!

Would there be any point in limiting the search to prime bases?

Bob

Check out [url]http://www.worldofnumbers.com/[/url]

Thanks
 

Thanks to Rogue for the reference to repunits in base 10. The list of repunit PRIMES is quite small. Only five have been discovered so far. In base 2 we have found 40 (?). At this very moment an elite world wide network of computers is almost continuously crunching numbers just to find the next Repunit in base two. In all probability yours is among them, dedicated to this task between your keystrokes.

The repunit primes in other bases were investigated by Stewart but i find no mirror site and his compilations may be stored deep in the computers of secretive mathematicians who delight in clandestine arcane manipulations far from the prying eyes of the internet and thus lost to civilization. Alas! Can we really trust Them to share??

Think how many centuries passed between the 13 polyhedra first described by Archimedes and their rediscovery by Kepler! Think how the value of Pi was calculated accurately to 15 decimals and then..and then... Lost..never to recover such accuracy again until the late 19th century .Need we revisit the Dark Ages when we, of all people, are keenly aware of the power of distributed computing at our disposal? Do we not owe it to ourselves and to the generations that follow to hold back the twilight of Mathematical indifference?


I Implore you,fellow searchers, to GENERALIZE. YOur noble efforts are indeed appreciated but there are so many of us focused exclusively on Base two repunits--it is as if All the other numbers do not exist! --surely in your generosity a few of you could dedicate a token of your time and talent to establish a DISTRIBUTED search for Repunits in other bases.

In base two the GRU can be tested using the n+1 test easily, but in general the main obstacle of GRU primality proving is to FACTOR the cyclotomic numbers involved in the factors of n-1.

GENERALIZED REPUNIT PRIMES
 

And so, wpolly, is there now any tabulation of those factors anywhere on the net?

[QUOTE=Bob Underwood]And so, wpolly, is there now any tabulation of those factors anywhere on the net?[/QUOTE]

Try here:

[url]http://mathworld.wolfram.com/Repunit.html[/url]

In general, a repunit in base [I]b[/I] is a number of the form

M[sup]b[/sup][sub]n[/sub] = (b[sup]n[/sup]-1)/(b-1)

It's easy to create a list of those numbers, but not trivial, according to their size, to check for their primality.


Luigi

I followed ET's link, and then to [URL=http://mathworld.wolfram.com/CunninghamNumber.html]Cunningham Number[/URL], at the bottom of the page it says[QUOTE]Updated factorizations were published in Brillhart et al. (1988). The tables have been extended by Brent and te Riele (1992) to
b = 13, ..., 100 with n < 255 for b < 30, and n < 100 for b >= 30.
All numbers with exponent 58 and smaller, and all composites with <= 90 digits have now been factored. :bow: [/QUOTE] So any coordinated effort should start somewhere beyond this.
To find any current efforts, I suggest you search for the Cunningham project, or for factors of Cunningham Numbers.

You could also check if Brent and te Riele published their tables.

I would really like to know what different algorithms were used for each of the searches. :geek:

[QUOTE=Maybeso]You could also check if Brent and te Riele published their tables.[/QUOTE]

Brent regularly updates a list of known factors of a[sup]n[/sup] ± 1 for a and n less than 10,000. It's pretty sparse for the larger values of both a and n, but gives a place to accumulate and share found factors. He doesn't appear to have any way to tell primes from unfactored composites, though.

[url]http://web.comlab.ox.ac.uk/oucl/work/richard.brent/factors.html[/url]

William

[QUOTE=Bob Underwood] The repunit primes in other bases were investigated by Stewart but i find no mirror site and his compilations may be stored deep in the computers of secretive mathematicians who delight in clandestine arcane manipulations far from the prying eyes of the internet and thus lost to civilization. Alas! Can we really trust Them to share??[/QUOTE]
In my experience, people who search for these things are only too willing to share. Other than the bragging rights for having found them, there are very, very few rewards for spending the time and effort involved in the search.

Note that all repunits are of the form (b^n-1)/(b-1) and so if you want to find examples in other people's results you should concentrate on such as the Cunningham project and Richard Brent's compendium of such things.

Paul

[QUOTE=Bob Underwood]And so, wpolly, is there now any tabulation of those factors anywhere on the net?[/QUOTE]
[color=#008000][url="http://www.asahi-net.or.jp/~KC2H-MSM/cn/"]www.asahi-net.or.jp/~KC2H-MSM/cn/[/url][/color]

[QUOTE=xilman]In my experience, people who search for these things are only too willing to share. Other than the bragging rights for having found them, there are very, very few rewards for spending the time and effort involved in the search.

Note that all repunits are of the form (b^n-1)/(b-1) and so if you want to find examples in other people's results you should concentrate on such as the Cunningham project and Richard Brent's compendium of such things.

Paul[/QUOTE]


All said tongue firmly embeded in cheek! Of course we share , Gauss being a notable exception. This was written in the faint hope that hyped eloquence might be considered mildly amusing and also in the hope that the distributed network could be applied to this search. I am not a mathematician but somehow find this problem reasonably enchanting. Elsewhere in these forums it has been said that "we" do not need another factoring project". ((whoever "WE" is )). Am i alone in thinking that a few non mathematicians would be willing to lend their computers to such a project? What could possibly be more fascinating than the number ONE? ( oops!, tongue in cheek -- Again!)

The beauty of GIMPS is that you professionals have finally allowed your cheering fans to touch the ball. Its a shame to see only a FEW powerful univerity computers crunching away at these calculations when entire networks of computing power remain idle. I work for the US Postal Service. Every supervisor has a computer and these computers are only used to retrieve files. If the Postal Service were to lend its vast network to your effort you can bet they would soon issue a commemorative stamp congratulating themselves as the proud discoverers of the 42nd .Of course ive suggested this to the powers that be...and have yet to recieve a reply.
Bob

See [URL]http://mersenneforum.org/showthread.php?t=21808[/URL]. For bases 2<=b<=1025 (perfect powers excluded), there are 56 bases have no known repunit primes (wth n>=3): 184, 185, 200, 210, 269, 281, 306, 311, 326, 331, 371, 380, 384, 385, 394, 396, 452, 465, 485, 511, 522, 570, 574, 598, 601, 629, 631, 632, 636, 640, 649, 670, 684, 691, 693, 711, 713, 731, 752, 759, 771, 795, 820, 861, 866, 872, 881, 932, 938, 948, 951, 956, 963, 996, 1005, 1015.

Now, there are only 32 bases <= 1024 without known repunit (probable) primes with length > 2: {185, 269, 281, 380, 384, 385, 394, 396, 452, 465, 511, 574, 598, 601, 629, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015}

Status text file attached.