## Re: [PrimeNumbers] primes such that every bit matters?

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• ... See http://oeis.org/A137985 and Terence Tao s paper (it mentions me!). The base-10 variant is called weakly prime numbers: http://oeis.org/A050249 -- Jens
Message 1 of 14 , Apr 3, 2013
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WarrenS wrote:
> A prime P which turns into a composite if you alter any bit in
> its binary representation is an "every bit matters" prime.

See http://oeis.org/A137985 and Terence Tao's paper (it mentions me!).
The base-10 variant is called weakly prime numbers: http://oeis.org/A050249

--
Jens Kruse Andersen
• thanks everybody (I actually knew most of that, but thanks anyhow)... I had not known about William Paulsen. Paulsen apparently conjectured that 67607 is the
Message 2 of 14 , Apr 3, 2013
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thanks everybody (I actually knew most of that, but thanks anyhow)...
that 67607 is the least prime such that changing any bit (including leading 0s)
always yields a composite(?). This would improve a lot versus 2131099

Unfortunately for his conjecture,
67607 + 2^16389
is prime (says MAPLE9 -- is it right?).
This prime shows as a side effect of proposition 2 in Paulsen's article that 67607 is not a Sierpinski number (or if it is, not one having a proof based on covering congruences)
which I think was not previously known.

Wm Paulsen: the prime numbers maze, Fibonacci Quart. 40 (2002) 272-279.
http://www.fq.math.ca/Scanned/40-3/paulsen.pdf

Paulsen also notes a candidate is 19249.
However, 19249*2^13018586+1 is prime
which defeats his argument although the conjecture per se is not refuted (yet).
That is, we know 19249 is not a Sierpinski number, and hence there can be no proof
based on covering congruences that 19249+2^k is always composite.
Hence it is plausible there exists a prime 19249+2^k (although I do not know one).

We can indeed kill quite a few Sierpinski candidates from the page
http://en.wikipedia.org/wiki/Seventeen_or_Bust
in the same way:

* 10223 + 2^k is prime if k=19, 103, or 3619
hence 10223 is not Sierpinski-via-covering-congruences.

* 21181 + 2^k is prime for k=28, 196, 268, and 316
hence 21181 is not Sierpinski-via-covering-congruences.

* 22699 + 2^k is prime for k=26 and 1250
hence is not Sierpinski-via-covering-congruences.

* 24737 + 2^k is prime for k=17
hence is not Sierpinski-via-covering-congruences.

* 55459 + 2^k is prime for k=14, 746, and 854
hence is not Sierpinski-via-covering-congruences.

This in fact kills every undecided case in the "seventeen or bust" project -- none
of them are Sierpinski-via-covering-congruences.
It is still possible they could be Sierpinski for some other reason (i.e. luck), though.

Paulsen also notes the prime bit-alteration graph is bipartite,
the "parity" of a prime (number of 1s in its binary representation is even or odd?)
governs that...
• ... Probably. PFGW says it is Fermat (to bases 3 & 137) and Lucas PRP, which (ducking a 16389-bit PRIMO proof) is good enough, I reckon. Mike
Message 3 of 14 , Apr 4, 2013
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--- In primenumbers@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
> thanks everybody (I actually knew most of that, but thanks anyhow)...
> that 67607 is the least prime such that changing any bit (including leading 0s)
> always yields a composite(?). This would improve a lot versus 2131099
> (which he also knew about).
>
> Unfortunately for his conjecture,
> 67607 + 2^16389
> is prime (says MAPLE9 -- is it right?).

Probably.
PFGW says it is Fermat (to bases 3 & 137) and Lucas PRP, which (ducking a 16389-bit PRIMO proof) is good enough, I reckon.

Mike
• ... It is surely unlikely that such an illustrious project has got it all wrong! The mistake is that all your remarks are about the so-called /dual/ Sierpinski
Message 4 of 14 , Apr 4, 2013
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--- In primenumbers@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
> thanks everybody (I actually knew most of that, but thanks anyhow)...
> that 67607 is the least prime such that changing any bit (including leading 0s)
> always yields a composite(?). This would improve a lot versus 2131099
> (which he also knew about).
>
> Unfortunately for his conjecture,
> 67607 + 2^16389
> is prime (says MAPLE9 -- is it right?).
> This prime shows as a side effect of proposition 2 in Paulsen's article that 67607 is not a Sierpinski number (or if it is, not one having a proof based on covering congruences)
> which I think was not previously known.
>
> Wm Paulsen: the prime numbers maze, Fibonacci Quart. 40 (2002) 272-279.
> http://www.fq.math.ca/Scanned/40-3/paulsen.pdf
>
> Paulsen also notes a candidate is 19249.
> However, 19249*2^13018586+1 is prime
> which defeats his argument although the conjecture per se is not refuted (yet).
> That is, we know 19249 is not a Sierpinski number, and hence there can be no proof
> based on covering congruences that 19249+2^k is always composite.
> Hence it is plausible there exists a prime 19249+2^k (although I do not know one).
>
> We can indeed kill quite a few Sierpinski candidates from the page
> http://en.wikipedia.org/wiki/Seventeen_or_Bust
> in the same way:
>
> * 10223 + 2^k is prime if k=19, 103, or 3619
> hence 10223 is not Sierpinski-via-covering-congruences.
>
> * 21181 + 2^k is prime for k=28, 196, 268, and 316
> hence 21181 is not Sierpinski-via-covering-congruences.
>
> * 22699 + 2^k is prime for k=26 and 1250
> hence is not Sierpinski-via-covering-congruences.
>
> * 24737 + 2^k is prime for k=17
> hence is not Sierpinski-via-covering-congruences.
>
> * 55459 + 2^k is prime for k=14, 746, and 854
> hence is not Sierpinski-via-covering-congruences.
>
> This in fact kills every undecided case in the "seventeen or bust" project -- none
> of them are Sierpinski-via-covering-congruences.
> It is still possible they could be Sierpinski for some other reason (i.e. luck), though.

It is surely unlikely that such an illustrious project has got it all wrong!

The mistake is that all your remarks are about the so-called /dual/ Sierpinski problem.

And in fact the Five or Bust website
http://www.mersenneforum.org/forumdisplay.php?f=86
tells us that 67607 + 2^16389 is indeed a /proven/ prime.

Mike
• ... --didn t claim they were wrong. And in fact, W.Keller pointed out to me this paper #A61 here: http://www.integers-ejcnt.org/vol8.html which would seem to
Message 5 of 14 , Apr 4, 2013
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> It is surely unlikely that such an illustrious project has got it all wrong!

--didn't claim they were wrong. And in fact, W.Keller pointed out to me this paper
#A61 here: http://www.integers-ejcnt.org/vol8.html
which would seem to confirm what I said (they already knew it).
It's just a bit weird that I thought I had a totally original problem, and it turns out it
has been worked on a ton by others for years...

Also, this frightening number is a probable prime:
19249+2^551542

> The mistake is that all your remarks are about the so-called /dual/ Sierpinski problem.
--which is... what?
• ... Section 2 of the paper to which Wilfrid directed you explains the diffrence between the SierpiĀ“nski problem and its dual, as remarked upon by Mike. David
Message 6 of 14 , Apr 4, 2013
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"WarrenS" <warren.wds@...> wrote:

> > The mistake is that all your remarks are about the so-called /dual/ Sierpinski problem.
> --which is... what?

Section 2 of the paper to which Wilfrid directed you
explains the diffrence between the SierpiĀ“nski problem
and its dual, as remarked upon by Mike.

David (atonally)
• ... Why might that seem weird to you, Warren? None of us should presuppose a monopoly on originality. Please see
Message 7 of 14 , Apr 4, 2013
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"WarrenS" <warren.wds@...> wrote:

> It's just a bit weird that I thought I had a totally original
> problem, and it turns out it has been worked on a ton by others
> for years...

Why might that seem "weird" to you, Warren?
None of us should presuppose a monopoly on originality.

http://primes.utm.edu/primes/page.php?id=110402
> Kaiser1, Broadhurst, OpenPFGW, NewPGen, Primo
for a laborious ECPP proof of a prime relevant to
the dual Sierpi'nski problem:
http://oeis.org/A076336/a076336c.html
> 21661 61792 Broadhurst [May 20, 2002]

In this case, neither Peter Kaiser nor I claim originality,
which is indeed a scarce commodity.

David
• ... FWIW, the pages are still available at http://web.archive.org/http://www.csm.astate.edu/~wpaulsen/primemaze/pmaze.html Maximilian
Message 8 of 14 , Apr 4, 2013
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> And although Paulsen's links seem to be dead, here's a message from
> 10+ years ago, to this very mailing list, offering up the number 2131099:

FWIW, the pages are still available at
http://web.archive.org/http://www.csm.astate.edu/~wpaulsen/primemaze/pmaze.html

Maximilian
• ... and is dwarfed by http://www.primenumbers.net/prptop/detailprp.php?rank=1 ... David
Message 9 of 14 , Apr 4, 2013
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"WarrenS" <warren.wds@...> wrote:

> this frightening number is a probable prime:
> 19249+2^551542

and is dwarfed by
> 2^9092392+40291

David
• ... If you paste this into OEIS (and probably google, too) you will immediately find A137985 which in the first comment links to A065092, which in turn
Message 10 of 14 , Apr 5, 2013
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>
> A prime P which turns into a composite if you alter any bit in
> its binary representation is an "every bit matters" prime.
>
> The examples below 10000 are
> 127, 173, 191, 223, 233, 239, 251, 257, 277, 337, 349, 373, 431, 443,
> ...

If you paste this into OEIS (and probably google, too)
you will immediately find A137985 which in the first comment links
to A065092, which in turn refers to Paulsen's Prime Numbers Maze.

Regards,
Maximilian
• ... There is something weird though - and that s that huge quantities of stuff I looked at a decade ago is being rediscovered by Warren. This makes my
Message 11 of 14 , Apr 9, 2013
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--- On Thu, 4/4/13, djbroadhurst wrote:
> "WarrenS" <warren.wds@...> wrote:
> > It's just a bit weird that I thought I had a totally original
> > problem, and it turns out it has been worked on a ton by others
> > for years...
>
> Why might that seem "weird" to you, Warren?
> None of us should presuppose a monopoly on originality.

There is something weird though - and that's that huge quantities of
stuff I looked at a decade ago is being rediscovered by Warren. This
makes my retirement from the field very hard, as he keeps posting
things that I've been directly interested in. However, I'm happy, as
a fresh mind approaching a problem can only ever increase the amount
that is known, never diminish it. In particular, whilst my arithmetic
may have been efficient, I was rarely good at the hard maths, so
hopefully Warren can get past the road-blocks that I had way back when.

Phil
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