- 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. - --- On Thu, 4/4/13, djbroadhurst wrote:
> "WarrenS" <warren.wds@...> wrote:

There is something weird though - and that's that huge quantities of

> > 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.

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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