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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}]}]
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