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

*To*: mathgroup at smc.vnet.net*Subject*: [mg97113] Re: Conjecture: 2n+1= 2^i+p ; 6k-2 or 6k+2 = 3^i+p*From*: Tangerine Luo <tangerine.luo at gmail.com>*Date*: Thu, 5 Mar 2009 04:57:42 -0500 (EST)*References*: <goj2cu$2s5$1@smc.vnet.net> <golr5o$qg0$1@smc.vnet.net>

Thanks! Because the moderator kindly told me that this group doesn't discuss general math questions, I posted it on http://groups.google.com/group/sci.math/browse_thread/thread/78fbaeb4c16154= ea# To my surprise, this question had been mentioned long years ago. In 1950 , a mathematician proved the conjecture is false! On 3=D4=C24=C8=D5, =CF=C2=CE=E78=CA=B111=B7=D6, "Sjoerd C. de Vries" <sjoer= d.c.devr... at gmail.com> wrote: > While I think questions like these may be great fun, I feel that they > don't belong in this forum as long as they are not connected with > Mathematica. I tried the brute force method for your particular > example of 2293 and the formal approach for the general conjecture. > > By brute force using > > Monitor[ > While[! PrimeQ[2^i - 2293], i++], > i > ] > > for 2 n +1 ==2293 I find after a long wait that i must be larger tha= n > 43 032 if the conjecture is correct. > > A formal way of stating the problem in Mathematica would be > > ForAll[n, n \[Element] Integers, > Exists[{i, p}, i \[Element] Integers \[And] p \[Element] Primes, > 2 n + 1 == 2^i + p \[Or] 2 n + 1 == 2^i - p]] > > Resolve or FullSimplify could then be used to find out whether or not > there is any truth in this statement. Alas, they both return unsolved. > > Cheers -- Sjoerd > > On Mar 3, 12:56 pm, Tangerine Luo <tangerine.... at gmail.com> wrote: > > > 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 =A3=ACI don't know i , p . it is sure that i>30 00= 0 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?