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Prime-go-round (Posted on 2012-02-06) Difficulty: 3 of 5
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).

See The Solution Submitted by Ady TZIDON    
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Solution computer solution | Comment 1 of 2

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

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