## Reflective and Cyclic Sets of Primes

```If the binary representations of the first several primes are reversed,
the resulting number is also irreducible (i.e., either a prime or 1).
This was pointed out by John Rickert, who cited the following
reflective pairs of primes

reverse prime
prime                (or unit)
2     10        01   1       (or 010 to 010)
3     11        11   3
5    101       101   5
7    111       111   7
11  1011      1101  13
13  1101      1011  11
17 10001     10001  17

The first prime whose binary reversal has a proper factorization is
19 = 10011, which gives 11001=25.  Still, of the primes less than
2048, more than half of them have prime reflective partners.

I suspect the proportion of such primes becomes arbitrarily small
as we go to larger and larger numbers.  To illustrate, here's a
little table showing the number of such primes less than x for
various values of x:

number of primes < x
total number      that give primes under
x     of primes < x     binary reversal         ratio
--------   -------------    ---------------------     -------
100          25               21                 0.8400
1000         168              102                 0.6071
10000        1229              509                 0.4141
100000        9592             3054                 0.3184
1000000       78498            20054                 0.2555
10000000      664579           141772                 0.2133

I would expect that the fraction of primes whose reflective pairs
are also primes is roughly what would be expected based on a random
selection of an odd number with the correct number of binary digits
for each prime.  Thus it becomes increasingly less likely as the
density of primes (slowly) decreases.  Also, there are presumably
infinitely many such prime pairs, although I don't know how to
prove it.

One interesting approach to this problem might be based on interpreting
the binary representations as polynomials with reciprocal roots,
as discussed in The Fundamental Theorem For Palindromic Polynomials.

By the way, here's some C code for counting reflective pairs:

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
unsigned long ic,d,nmax, nn, n, i, imax, mm, kk, aa[30];
unsigned long ictot,ptot;
main ()
{
nmax=1000;
for(nn=3;nn<=nmax;nn=nn+2)
{
ic=0;for(d=3;d<sqrt(nn+1);d=d+2)if(nn%d==0)ic=1;
if(ic==0)
{
for(i=0;i<31;i++)aa[i]=0;
n=nn;i=0;
while(n>0){aa[i]=n%2;n=(n-aa[i])/2;i++;}imax=i-1;
mm=0;for(i=0;i<=imax;i++)mm=2*mm+aa[i];
ic=0;for(d=3;d<sqrt(mm+1);d=d+2)if(mm%d==0)ic=1;
ictot=ictot+ic; ptot=ptot+1;
printf(" %lu %lu %lu %lu %lu \n",ptot,nn,mm,ic,ictot);
}
}
}

On a slightly related point, we can also consider "complete cyclic
sets of primes", i.e., primes whose base-b expansions are primes
under all the rotations of their digits.  Obviously there are none
in the base 2 other than repunits, i.e., primes of the form 2^p - 1.
However, for any higher base there exist complete cyclical prime
sets.  Focusing on the base 10, some trivial examples of two-digit
cyclic pairs of primes are

11        13       37       17       79
11        31       73       71       97

For slightly less trivial examples, we have the three-digit cyclical
prime sets

113       197       199      337
311       719       919      733
131       971       991      373

Among four-digit decimal numbers the only cyclic sets of primes are

1193          3779
3119          9377
9311          7937
1931          7793

Likewise among the five-digit decimal numbers we have only the sets

11939         19937
91193         71993
39119         37199
93911         93719
19391         99371

Finally, among six-digit decimal numbers the only cyclic sets of
primes are
193939        199933
919393        319993
391939        331999
939193        933199
393919        993319
939391        999331

There are no complete cyclic sets of decimal primes with 7 or 8 digits.
The next known cycle set is the repunit with 19 digits, then the repunits
with 23, 317, 1031 digits, and so on. If we exclude repunits, it is not
known if there are any complete cyclic sets of primes in the base 10 with
more than 6 digits.

Of course, we can also find cyclic sets of primes in other bases.  For
example, in the base 7 we have the set

11515
51151
15115
51511
15151

Also, whenever a repunit is a prime, it gives a degenerate cyclic set,
such as the number 1111111 in the base 3 (which equals the prime 1093
in decimal).
```