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2-defensive prime number

• To: mathgroup at smc.vnet.net
• Subject: [mg98345] 2-defensive prime number
• From: Tangerine Luo <tangerine.luo at gmail.com>
• Date: Wed, 8 Apr 2009 02:46:44 -0400 (EDT)

2-defensive odd number is defined as:
n=2k+1 is an odd number, k>0, if Mod[2^i+1,n] != 0 for any natural
number i , then n is 2-defensive odd number. If n is prime too, then n
is 2-defensive prime number.

My questions are:
1. Have 2-defensive odd numbers some special form?
2. ratio = all 2-defensive odd numbers / all odd numbers >1/2 ?
    ratio = all 2-defensive prime numbers / all odd prime numbers <
1/2 ?


Euler's Theorem: If (a,m)=1, then m|a^EulerPhi[m] -1
so , m | a^(i+EulerPhi[m]) - a^i  for any natural number i.
therefor, if Mod[2^i+1,n] != 0 for i between [1, EulerPhi[n])] , then
n is 2-defensive odd number.

for example,
3|2^1+1

5|2^2+1

7 is 2 defensive prime number

9|2^3+1

11|2^5+1

13|2^6+1

15 is 2 defensive odd number
...
23 is 2 defensive prime number

My program code is:

Table[
 For[i = 1, i < p, i++,
      If[Mod[2^i + 1, p] == 0, Print[p, "|2^", i, "+1"]; Break[]]
   ]
  If[i == p, Print[p, " is 2 defensive number"]]
 ,
 {p, Table[Prime[i], {i, 100}]}]

• Follow-Ups:
  • Re: 2-defensive prime number
    • From: Daniel Lichtblau <danl@wolfram.com>