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>