For k=2,3,4,5,6 find an example (or prove absence) of a circular prime of k digits (i.e. a prime number that remains prime as its leftmost digit (Most Significant Digit) is moved in turn to the right hand side).

The lowest member of each family of circular primes is listed:

11
13
17
37
79
113
197
199
337
1193 
3779 
11939
19937
193939
199933

from

  10   P=7
  20   while P<100000000
  30     P=nxtprm(P)
  40     Ps1=cutspc(str(P))
  43     Good=1
  47     for I=1 to len(Ps1)-1
  50       Ps2=mid(Ps1,2,*)+left(Ps1,1)
  60       P2=val(Ps2):if P2<P then Good=0
  65       if prmdiv(P2)<P2 then Good=0:cancel for:goto 90
  70       Ps1=Ps2
  75     next
  80     if Good then print P
  90   wend
 
showing that 7- and 8-digit circular primes do not exist. 

A search for 113,197,199,337 in Sloane's OEIS finds sequence A016114, which starts with single digits (k=1). Google search finds that it's conjectured that beyond 199933 are only repunit primes. The primes listed by Sloane are [2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, 1111111111111111111, 11111111111111111111111].

Edited on February 6, 2012, 4:52 pm

Posted by Charlie on 2012-02-06 16:50:13