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primes such that every bit matters?

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WarrenS     Message 1 of 14  Apr 3, 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,
491, 509, 557, 653, 683, 701, 733, 761, 787, 853, 877, 1019, 1193,
1201, 1259, 1381, 1451, 1453, 1553, 1597, 1709, 1753, 1759, 1777,
1973, 2027, 2063, 2333, 2371, 2447, 2633, 2879, 2917, 2999, 3083,
3181, 3209, 3313, 3511, 3593, 3643, 3767, 3779, 3851, 3877, 3889,
3967, 4013, 4177, 4283, 4441, 4451, 4561, 4597, 4603, 4679, 4813,
4889, 4951, 5051, 5099, 5209, 5323, 5557, 5801, 5867, 6007, 6073,
6151, 6203, 6211, 6287, 6323, 6379, 6481, 6521, 6971, 6977, 6997,
7027, 7039, 7043, 7103, 7109, 7151, 7207, 7297, 7307, 7331, 7369,
7507, 7573, 7583, 7841, 7883, 8017, 8087, 8111, 8171, 8231, 8243,
8311, 8363, 8627, 8747, 8831, 8849, 8867, 8923, 9137, 9151, 9161,
9319, 9323, 9697, 9767

The Mersenne primes P=2^p-1 also have this "every bit matters" property when
p =  7, 31, 127, 607, 1279, 4423
for the p<10000.

My current conjecture is that a fraction B of all primes are every-bit-matters primes,
where B = exp(-2*C2 / ln2) = 0.14884878474999065378100135978
where
C2=0.660161815846869573927812110014...
is the Hardy Littlewood twin prime constant described here
http://en.wikipedia.org/wiki/Twin_prime#First_Hardy.E2.80.93Littlewood_conjecture
[This B agrees mildly well with my computer counts for primes<10^9.
If you count among the primes up to N I think the error in B will be of order 1/logN,
so convergence expected to be slow.  But perhaps with extrapolate-to-infinity tricks
you could get more convincing evidence confirming or denying the conjecture.]

It also is interesting to ask: what if the infinity of leading 0s are also considered "bits"
susceptible to alteration?
The following prime
2131099 = ...0000000000001000001000010010011011 binary
has the property that if any bit is altered, you get a composite.
Is this the least such prime?  I do not know.

I can prove by stealing a result of Zhi-Wei Sun 
that liminfB >= 1/9761888869657922764800000000
(and this still works even for the "leading 0s allowed" version)
but have no proof that limsupB<1 or that limB exists.
===============================================
Jack Brennen     Message 2 of 14  Apr 3, 2013
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See here for something related...

http://www.primepuzzles.net/problems/prob_025.htm


 On 4/3/2013 7:06 PM, 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.
>
> The examples below 10000 are
> 127, 173, 191, 223, 233, 239, 251, 257, 277, 337, 349, 373, 431, 443,
> 491, 509, 557, 653, 683, 701, 733, 761, 787, 853, 877, 1019, 1193,
> 1201, 1259, 1381, 1451, 1453, 1553, 1597, 1709, 1753, 1759, 1777,
> 1973, 2027, 2063, 2333, 2371, 2447, 2633, 2879, 2917, 2999, 3083,
> 3181, 3209, 3313, 3511, 3593, 3643, 3767, 3779, 3851, 3877, 3889,
> 3967, 4013, 4177, 4283, 4441, 4451, 4561, 4597, 4603, 4679, 4813,
> 4889, 4951, 5051, 5099, 5209, 5323, 5557, 5801, 5867, 6007, 6073,
> 6151, 6203, 6211, 6287, 6323, 6379, 6481, 6521, 6971, 6977, 6997,
> 7027, 7039, 7043, 7103, 7109, 7151, 7207, 7297, 7307, 7331, 7369,
> 7507, 7573, 7583, 7841, 7883, 8017, 8087, 8111, 8171, 8231, 8243,
> 8311, 8363, 8627, 8747, 8831, 8849, 8867, 8923, 9137, 9151, 9161,
> 9319, 9323, 9697, 9767
>
> The Mersenne primes P=2^p-1 also have this "every bit matters" property when
>     p =  7, 31, 127, 607, 1279, 4423
> for the p<10000.
>
> My current conjecture is that a fraction B of all primes are every-bit-matters primes,
> where B = exp(-2*C2 / ln2) = 0.14884878474999065378100135978
> where
>   C2=0.660161815846869573927812110014...
> is the Hardy Littlewood twin prime constant described here
> http://en.wikipedia.org/wiki/Twin_prime#First_Hardy.E2.80.93Littlewood_conjecture
> [This B agrees mildly well with my computer counts for primes<10^9.
> If you count among the primes up to N I think the error in B will be of order 1/logN,
> so convergence expected to be slow.  But perhaps with extrapolate-to-infinity tricks
> you could get more convincing evidence confirming or denying the conjecture.]
>
> It also is interesting to ask: what if the infinity of leading 0s are also considered "bits"
> susceptible to alteration?
> The following prime
>    2131099 = ...0000000000001000001000010010011011 binary
> has the property that if any bit is altered, you get a composite.
> Is this the least such prime?  I do not know.
>
> I can prove by stealing a result of Zhi-Wei Sun
> that liminfB >= 1/9761888869657922764800000000
> (and this still works even for the "leading 0s allowed" version)
> but have no proof that limsupB<1 or that limB exists.
>
> 
===============================================
Jack Brennen     Message 3 of 14  Apr 3, 2013
-----------------------------------------------
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:

http://tech.groups.yahoo.com/group/primenumbers/message/7301


 On 4/3/2013 7:20 PM, Jack Brennen wrote:
> See here for something related...
>
>      http://www.primepuzzles.net/problems/prob_025.htm
>
>
>
> On 4/3/2013 7:06 PM, 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.
>>
>> The examples below 10000 are
>> 127, 173, 191, 223, 233, 239, 251, 257, 277, 337, 349, 373, 431, 443,
>> 491, 509, 557, 653, 683, 701, 733, 761, 787, 853, 877, 1019, 1193,
>> 1201, 1259, 1381, 1451, 1453, 1553, 1597, 1709, 1753, 1759, 1777,
>> 1973, 2027, 2063, 2333, 2371, 2447, 2633, 2879, 2917, 2999, 3083,
>> 3181, 3209, 3313, 3511, 3593, 3643, 3767, 3779, 3851, 3877, 3889,
>> 3967, 4013, 4177, 4283, 4441, 4451, 4561, 4597, 4603, 4679, 4813,
>> 4889, 4951, 5051, 5099, 5209, 5323, 5557, 5801, 5867, 6007, 6073,
>> 6151, 6203, 6211, 6287, 6323, 6379, 6481, 6521, 6971, 6977, 6997,
>> 7027, 7039, 7043, 7103, 7109, 7151, 7207, 7297, 7307, 7331, 7369,
>> 7507, 7573, 7583, 7841, 7883, 8017, 8087, 8111, 8171, 8231, 8243,
>> 8311, 8363, 8627, 8747, 8831, 8849, 8867, 8923, 9137, 9151, 9161,
>> 9319, 9323, 9697, 9767
>>
>> The Mersenne primes P=2^p-1 also have this "every bit matters" property when
>>      p =  7, 31, 127, 607, 1279, 4423
>> for the p<10000.
>>
>> My current conjecture is that a fraction B of all primes are every-bit-matters primes,
>> where B = exp(-2*C2 / ln2) = 0.14884878474999065378100135978
>> where
>>    C2=0.660161815846869573927812110014...
>> is the Hardy Littlewood twin prime constant described here
>> http://en.wikipedia.org/wiki/Twin_prime#First_Hardy.E2.80.93Littlewood_conjecture
>> [This B agrees mildly well with my computer counts for primes<10^9.
>> If you count among the primes up to N I think the error in B will be of order 1/logN,
>> so convergence expected to be slow.  But perhaps with extrapolate-to-infinity tricks
>> you could get more convincing evidence confirming or denying the conjecture.]
>>
>> It also is interesting to ask: what if the infinity of leading 0s are also considered "bits"
>> susceptible to alteration?
>> The following prime
>>     2131099 = ...0000000000001000001000010010011011 binary
>> has the property that if any bit is altered, you get a composite.
>> Is this the least such prime?  I do not know.
>>
>> I can prove by stealing a result of Zhi-Wei Sun
>> that liminfB >= 1/9761888869657922764800000000
>> (and this still works even for the "leading 0s allowed" version)
>> but have no proof that limsupB<1 or that limB exists.
>>
===============================================
Jens Kruse Andersen     Message 4 of 14  Apr 3, 2013
-----------------------------------------------
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
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WarrenS     Message 5 of 14  Apr 3, 2013
-----------------------------------------------
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 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.

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...
===============================================
mikeoakes2     Message 6 of 14  Apr 4, 2013
-----------------------------------------------
--- In primenumbers@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
> 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 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
===============================================
mikeoakes2     Message 7 of 14  Apr 4, 2013
-----------------------------------------------
--- In primenumbers@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
> 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 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
===============================================
WarrenS     Message 8 of 14  Apr 4, 2013
-----------------------------------------------
> 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?
===============================================
djbroadhurst     Message 9 of 14  Apr 4, 2013
-----------------------------------------------
--- In primenumbers@yahoogroups.com, 
"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 Sierpi´nski problem 
and its dual, as remarked upon by Mike.

David (atonally)
===============================================
djbroadhurst     Message 10 of 14  Apr 4, 2013
-----------------------------------------------
--- In primenumbers@yahoogroups.com, 
"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.

Please see 
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
===============================================
Maximilian Hasler     Message 11 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
===============================================
djbroadhurst     Message 12 of 14  Apr 4, 2013
-----------------------------------------------
--- In primenumbers@yahoogroups.com, 
"WarrenS" <warren.wds@...> wrote:

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

and is dwarfed by 
http://www.primenumbers.net/prptop/detailprp.php?rank=1
> 2^9092392+40291

David
===============================================
Maximilian Hasler     Message 13 of 14  Apr 5, 2013
-----------------------------------------------
>
> 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
===============================================
Phil Carmody     Message 14 of 14  Apr 9, 2013
-----------------------------------------------
--- 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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