Existence of a prime with some additional properties

Question

Does there exist a prime $p$ such that $p-1$ is squarefree, divisible by at least three primes, and $$ \{1^{\sigma(1)},\ldots,(2p)^{\sigma(2p)}\}=\{1,\ldots,2p\} $$ in $\mathbf{Z}/(2p)\mathbf{Z}$ for some permutation $\sigma$ of $\{1,\ldots,2p\}$?

I am asking only about numerical evidences: I conjecture the answer is negative, but I would be happy with a (counter)example as well.. For instance, does $p=31$ work?

[The question comes from a characterization of special type of primes which have a number of applications, e.g., in cryptography and primality testing. The motivation for this question is related to this thread and this problem.]

@JoshuaLochner You mean, checking randomly permutations of (at least) 62 elements? [...] — Paolo Leonetti, Aug 19 '16 at 14:38
Because $p=31$ is the smallest such number, and you want permutations of $2p$ elements. — rogerl, Aug 19 '16 at 14:51
@Paolo Not sure this helps, but note that under the implicit map from $\mathbb{Z}/(2p)\mathbb{Z}\to \mathbb{Z}/(2p)\mathbb{Z}$ defined by any fixed $\sigma$, the unit group of $\mathbb{Z}/(2p)\mathbb{Z}$ (which has $p-1$ elements) is mapped to itself. — rogerl, Aug 19 '16 at 14:59
Because the unit group is a group, so any power of an element of that group is again an element. I just realized perhaps my first comment was unclear. I didn't mean to imply that it was mapped onto itself, just into. — rogerl, Aug 19 '16 at 15:15
@rogerl Indeed I understood that $\sigma(x)$ would be a unit for every unit $x$, but probably it is false.. — Paolo Leonetti, Aug 19 '16 at 18:26

score 1 · Accepted Answer · 2016-08-24T14:16:07.500

1

$p = 31$

octave/matlab format:

sigma = [30   48   41   12   50    5   14    4   62   58   19   17   49   46   57   54   29   56   52   26   11    7   27   47   20   35   51   34   39   15   60   42   24    1   18   40   25   32   36   22  38   53   43   23   44   33    8   31   16    2   28   13   59   21   37   10   55    3   61    9   45    6];

sigma = [ 30   48   41   12   50    5   14    4   62   58 ...
          19   17   49   46   57   54   29   56   52   26 ...
          11    7   27   47   20   35   51   34   39   15 ...
          60   42   24    1   18   40   25   32   36   22 ...
          38   53   43   23   44   33    8   31   16    2 ...
          28   13   59   21   37   10   55    3   61    9 ...
          45    6];

Octave was used to generate Boolean equations in CNF format.

zChaff was used to solve Boolean selection variables.

More details:

A power residue matrix was created $R(r,c) = r^c \pmod {2p}$ for $r,c = 1 \dots 2p$

A Boolean decision matrix can be imagined where $D(r,c) = \delta_{r,c}$ where $\delta_{r,c} \in \mathbb{B}$ Boolean so that $\sigma = [1:2p]D^T$.

The conditions on $\delta_{r,c}$ are that exactly one is selected from each row and each column.

For each residue $0 \dots (2p-1)$ only one of the $\delta_{r,c}$ corresponding to the positions of that residue in $R$ will be true.

i.e. create Boolean variables to select only one of each residue from $R$.

edited Aug 24 '16 at 14:16

answered Aug 24 '16 at 13:50

I tried $p = 43$ but zchaff ran out of memory on my machine – Aug 25 '16 at 02:11
matlab mod exponential function: https://gist.github.com/ttezel/4635562 – Aug 25 '16 at 02:55
This is what I was searching for, thank you. Is there no way to check it also for $p=43$? I could add you some simplifications: if $\sigma(x)$ is even with $x \neq {1,p,p+1,2p}$ then $x$ is a quadratic residue; conversely, if $x \neq {1,p,p+1,2p}$ is a quadratic non-residue then $\sigma(x)$ is odd. [Note that $x^{\sigma(x)}\equiv x\pmod{2p}$ for each $x \in {1,p,p+1,2p}$] – Paolo Leonetti Aug 25 '16 at 08:18
@Paolo Leonetti : zchaff ran out of memory on an 8GByte machine after running for 8 hrs. The number of variables is $(2p)^2$. I've never used GIT Hub but I could try to put all the source code there and give the location in a comment. – Aug 25 '16 at 11:23
@Paolo Leonetti : Added the files to GITHUB : https://github.com/arthur67/Existence-of-a-prime-with-some-additional-properties See Instructions.txt g86.cnf is the file that needs to be solved by any logic solver – Aug 25 '16 at 12:34
@Paolo Leonetti : If you run zchaff on a UNIX operating system it could use the disk drive for memory swapping depending on the UNIX configuration. This would allow zchaff to work on larger problems e.g. $p = 43$. I use cygwin, a UNIX emulator over Windows 7 that won't swap memory to disk to the best of my knowledge. In UNIX look at the command ulimit to set unlimited resources for the shell. – Aug 25 '16 at 17:18
Thanks for the advices and the uploaded codes, I will try it during the weekend when I come back home – Paolo Leonetti Aug 25 '16 at 19:53

score 0 · Answer 2 · answered Sep 10 '16 at 19:30

This is not an answer, just an idea. Let o(x) denote the multiplicative order of x in Z*/(31)and let s be sigma. Note that if o(i^s(i)) = 30, then o(i) =30. Consider i: 55 53 11 17 43 21 13 3 and i^s(i): 3 55 53 11 17 43 21 13 where i+1 is the j for which j^s(j)=i The rows are just permutations of the phi(30)=8 members of Z*/(31) with o(x)=30. To find a permutation for n=2(43) maybe you can first determine what works for the 12=phi(42) members of Z*/(43) with order 42. I'm GUESSING 2(43) will be nice but 2(71) will be naughty. Please let me know if that guess turns out to be right.

Existence of a prime with some additional properties

2 Answers2

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