- 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... - --- 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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