# primes in pi: 1, 2, 6, 38, 16208, 47577, 78073, ...

Jonathan Post jvospost3 at gmail.com
Mon Jul 24 22:45:38 CEST 2006

```I am somewhat serious about pi base e.  So far as I can see, pi base e =
10.101002020002111120020101120... as per
A050948<http://www.research.att.com/~njas/sequences/A050948> Pi
expressed in base 1/e: Pi = Sum a(i)*exp(-i), i=-1,0,1,...
Now, in what base should we examine substrings of this for primality?  In
the unnatural base 10, we have the primes:
101, 101010020200021111, ...

The 62-digit string takes about a minute on the PC I'm using right now while
to factor:
10 101002 020002 111120 020101 120001 010202 000111 012020 010120 002001 =
409 x 773 197250 487261 611198 297287 x
31 941172 112561 420564 909184 938047

ahhh, and the 3rd prime is:

101010020200021111200201011200010102020001110120200101200020011011201012100100021001

Or, as the Alpertron puts it:

101010 020200 021111 200201 011200 010102 020001 110120 200101 200020 011011
201012 100100 021001
is prime

Anyone want to extend this, or verify or find I've erred?

-- Jonathan Vos Post
[in 3rd consectutive day with no electrical power to home in Southern
California during record heat wave and expolding transformers]

On 7/23/06, Russell Walsmith <ixitol at gmail.com> wrote:
>
> The idea of using the binary expansion of pi suggests another way to get a
> sequence of primes:
>
> Starting with the first digit of A004601 =
> 1,1,0,0,1,0,0,1,0,0,0,0,1,1,1,1,1,1,0,1,1,0,1,... we move rightward until we
> encounter another 1. Since 11 (= 3 in decimal) is prime, we move to the next
> 1 and repeat the process.
> 11 = 3
> 1001001 = 73
> 1001000011111 = 4639
> 1000011 = 67
> 11 = 3
> 11 = 3
> 11 = 3
>
> This gives the sequence 3, 73, 4639, 67, 3, 3, 3, 3, 3, 5, 3, 5, 5, 5, 17,
> 17, 1069...
>
> Can anyone extend this?
>
>
> Russell
>
>
>
>
> On 7/23/06, N. J. A. Sloane <njas at research.att.com> wrote:
> >
> > Richard Guy said:
> > > It would be more natural (??) to do this in base 2.
> > >
> > > So here are two new(?) sequences for people to
> > > check and extend:
> > >
> > > The first
> > >           2,  8,    14,    18,   ...
> > >
> > > digits in the binary representation of pi form the
> > > primes
> > >           3, 401, 25667, 410687, ...
> > >
> > > in decimal notation.
> >
> > Me:
> >
> > The binary expansion of Pi is A004601:
> > 1,1,0,0,1,0,0,1,0,0,0,0,1,1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,...
> >
> > Converting first n bits to an integer gives A068425:
> > 1,3,6,12,25,50,100,201,402,804,1608,3216,...
> >
> > The primes here are A117721:
> > 3,6588397,1686629713,26986075409,16703571626015105435307505830654230989,
> > ...
> >
> > and they occur for these values of n (A065987):
> > 2,23,31,35,124,323,2787,5717,6506 (and that's all I have)
> > The latter sequence was computed by Bob Wilson.
> >
> > Eric, can you extend it?
> >
> > Richard, I seem to disagree with your results, but perhaps
> > I misunderstood your message?
> >
> > Neil
> >
>
>
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