# [seqfan] Re: primes beginning 101 in binary

Neil Sloane njasloane at gmail.com
Sun Sep 20 17:32:01 CEST 2015

```Vladimir, Thank you for those two theorems of Sierpinski. That is very
helpful.  I edited A249974 a bit and approved it.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Email: njasloane at gmail.com

On Sun, Sep 20, 2015 at 8:15 AM, Vladimir Shevelev <shevelev at bgu.ac.il>
wrote:

> There are two old (1951) theorems by W. Sierpinsky (in the submitted
> to-day A249974 I gave two references):
> Theorem 1. If  c_1..c_m are decimal digits and c_1 is not zero,
> then there are infinitely many primes beginning from the sequence
> of the digits c_1..c_m.
> Theorem 2. If c_1..c_m are decimal digits such that c_m equals
> 1,3,7 or 9, then there are infinitely many primes ending on the
> sequence of the digits c_1..c_m.
> The proofs essentially do not depend on the base, i.e., 10 one can
> change by integer b>=2.
>
> I give these theorems by an inverse translation from Russian,
> from Russian issue of the book "Primzahlen" by Ernst Trost,
> Theorem 20-21.
>
> Best regards,
>
> ________________________________________
> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of David Wilson [
> davidwwilson at comcast.net]
> Sent: 19 September 2015 18:26
> To: 'Sequence Fanatics Discussion list'
> Subject: [seqfan] Re: primes beginning 101 in binary
>
> Yes, this type of sequence is infinite.
>
> By the prime number theorem, if eps > 0, there is a prime between n and
> n(1+e) for sufficient n.
> This implies that there is a prime between cb^k and (c+1)b^k for
> sufficient k.
> This means that there are an infinitude of numbers in base b starting with
> prefix c.
>
> > -----Original Message-----
> > From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Neil
> > Sloane
> > Sent: Saturday, September 19, 2015 10:11 AM
> > To: Sequence Fanatics Discussion list
> > Subject: [seqfan] primes beginning 101 in binary
> >
> > Alois has produced a clean binary version of A089755 and A262283, namely
> > A262350:
> > a(1) = 2. For n>1, let s denote the binary string of a(n-1) with the
> leftmost 1
> > and following consecutive 0's removed. Then a(n) is the smallest prime
> not
> > yet present whose binary representation begins with s.
> >
> > This led me to look at "primes beginning 101 in binary" etc.:
> > Primes whose binary expansion begins with binary expansion of 1, 2, 3,
> 4, 5,
> > 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
> >
> > My question is, is it known that any of these sequences are infinite
> (other
> > than A000040, the primes themselves)?
> >
> > Is there a proof that the new sequence A262350 or its complement are
> > infinite?
> >
> > _______________________________________________
> >
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>
>
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