# [seqfan] Re: primes beginning 101 in binary

Vladimir Shevelev shevelev at bgu.ac.il
Sun Sep 20 14:15:16 CEST 2015

```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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