## Primes from permutation of prime digits

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• Hi, prime folks, Primes from permutation of prime digits ======================================= For each prime p, define perm(p) = number of permutations of
Message 1 of 7 , Feb 11, 2009
Hi, prime folks,

Primes from permutation of prime digits
=======================================

For each prime p, define perm(p) =
number of permutations of digits of p which give primes
(including p itself).
Some examples:

perm(2)=1

perm(1097)=10: ten primes
{179, 197, 719, 971, 1097, 1709, 1907, 7019, 7109, 7901}
(note some primes arise from permutations with leading zeroes:
0179=>179, 0197=>197, etc.)

perm(1006991)=100: hundred primes
{11699,11969,19961,...9906101,9910601,9960101}

perm(12338449)=1000: thousand primes
{12338449, 12394483,12394843,...,98432413,98433241, 98433421}

We have the next table of
minimal primes p with given perm:
{perm,p)
{1,2}
{10,1097}
{100,1006991}
{1000,12338449}
{10000,??}
{100000,??}
any1 wish to extend this?
thx, zak
• ... first case of perm(p) 10^4 is perm(100123697)=10042 and of course i mean distinct primes , otherwise perm(11)=2 - curious enough. maximal perm for first
Message 2 of 7 , Feb 12, 2009
--- On Thu, 2/12/09, Maximilian Hasler <maximilian.hasler@...> wrote:

> From: Maximilian Hasler <maximilian.hasler@...>
> Subject: Re: [PrimeNumbers] Primes from permutation of prime digits
> To: zakseidov@...
> Date: Thursday, February 12, 2009, 8:40 AM
> --- In primenumbers@yahoogroups.com, zak seidov wrote:
> > For each prime p, define perm(p) =
> > number of permutations of digits of p which give
> primes
> > (including p itself).
> (...)
> > We have the next table of
> > minimal primes p with given perm:
> > {perm,p)
> > {1,2}
> > {10,1097}
> > {100,1006991}
> > {1000,12338449}
> > {10000,??}
>
> The definition does not correspond to the data above.
>
> zak(p,d,n)={sum(i=1,(n=#d=Vec(Str(p)))!,ispseudoprime(eval(concat(vecextract(d,numtoperm(n,i))))))}
>
> zak(1097)
> = 10
>
> zak(12338449)
> = 4000
>
> Indeed my function counts, according to your definition,
> the number of
> permutations of digits yielding primes, including those
> that just
> exchange two identical digits, etc.
>
> Whereas your data corresponds to the number of distinct
> primes
> produced (which may be smaller by some (generalized)
> binomial factor
> depending on the number of identical digits).
>
> Using your definition & my code the first with
> perm>10^4 I found
> (without exhaustive search) is:
> 11111117 15120
>
> Regards,
> Maximilian

first case of perm(p)>10^4 is perm(100123697)=10042
and of course i mean "distinct primes",
otherwise perm(11)=2 - curious enough.
maximal perm for first 2*10^7 primes is
perm(102345697)=30852 (distinct primes!)
zak
• ... Interesting. It s surprising to me that the number of odd digits barely outnumbers the number of even digits in these primes with maximum perms. Must be a
Message 3 of 7 , Feb 13, 2009
--- In primenumbers@yahoogroups.com, zak seidov <zakseidov@...> wrote:

> > > We have the next table of
> > > minimal primes p with given perm:
> > > {perm,p)
> > > {1,2}
> > > {10,1097}
> > > {100,1006991}
> > > {1000,12338449}
...
> > zak(12338449)
> > = 4000
> > Using your definition & my code the first with
> > perm>10^4 I found
> > (without exhaustive search) is:
> > 11111117 15120
...
> first case of perm(p)>10^4 is perm(100123697)=10042
> and of course i mean "distinct primes",
> otherwise perm(11)=2 - curious enough.
> maximal perm for first 2*10^7 primes is
> perm(102345697)=30852 (distinct primes!)
> zak
>

Interesting. It's surprising to me that the number of odd digits
barely outnumbers the number of even digits in these primes with
maximum perms. Must be a good reason ...

Mark
• ... Scrap that. It s not that surprising to me anymore. :) Mark
Message 4 of 7 , Feb 13, 2009
<mark.underwood@...> wrote:
>

> Interesting. It's surprising to me that the number of odd digits
> barely outnumbers the number of even digits in these primes with
> maximum perms. Must be a good reason ...
>

Scrap that. It's not that surprising to me anymore. :)

Mark
• I prefer zak(1123465789) = 152526 to zak(1023345679) = 156227 since the former does not invoke leading zeros. David
Message 5 of 7 , Feb 14, 2009
I prefer
zak(1123465789) = 152526 to
zak(1023345679) = 156227 since
the former does not invoke leading zeros.

David
• ... Thereafter, 11-digit primes become somewhat memory-intensive, in my simple-minded implementation, using GP s vecsort . Might Zak and/or Maximilian confirm
Message 6 of 7 , Feb 14, 2009

> I prefer
> zak(1123465789) = 152526 to
> zak(1023345679) = 156227 since
> the former does not invoke leading zeros.

Thereafter, 11-digit primes become somewhat memory-intensive,
in my simple-minded implementation, using GP's "vecsort".

Might Zak and/or Maximilian confirm that
zak(10123457689) = 1404250
without duplication of primes, but allowing Zak's leading zeros ?

David
• ... yes: zak(10123457689) %1 = 2808500 2808500/2 (symmetry factor for exchange of the two 1 s) %2 = 1404250 Maximilian
Message 7 of 7 , Feb 15, 2009
> Thereafter, 11-digit primes become somewhat memory-intensive,
> in my simple-minded implementation, using GP's "vecsort".
>
> Might Zak and/or Maximilian confirm that
> zak(10123457689) = 1404250
> without duplication of primes, but allowing Zak's leading zeros ?

yes:
zak(10123457689)
%1 = 2808500
2808500/2 \\ (symmetry factor for exchange of the two 1's)
%2 = 1404250

Maximilian
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