Conjecture: 2n+1= 2^i+p ; 6k-2 or 6k+2 = 3^i+p

*To*: mathgroup at smc.vnet.net*Subject*: [mg97037] Conjecture: 2n+1= 2^i+p ; 6k-2 or 6k+2 = 3^i+p*From*: Tangerine Luo <tangerine.luo at gmail.com>*Date*: Tue, 3 Mar 2009 05:56:33 -0500 (EST)

I have a conjecture: Any odd positive number is the sum of 2 to an i-th power and a (negative) prime. 2n+1 = 2^i+p for example: 5 = 2+3 9=4+5 15=2^3+7 905=2^12-3191 .... as to 2293=2^i +p ，I don't know i , p . it is sure that i>30 000 if the conjecture is correct. More, n = 3^i+p, (if n=6k-2 or n=6k+2) for example:8 = 3+5 16=3^2+7 100=3+97, 562 = 3^6 -167 I can't proof this. Do you have any idea?

**Follow-Ups**:**Re: Conjecture: 2n+1= 2^i+p ; 6k-2 or 6k+2 = 3^i+p***From:*Daniel Lichtblau <danl@wolfram.com>

**Re: Conjecture: 2n+1= 2^i+p ; 6k-2 or 6k+2 = 3^i+p***From:*DrMajorBob <btreat1@austin.rr.com>