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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg97119] Re: Conjecture: 2n+1= 2^i+p ; 6k-2 or 6k+2 = 3^i+p
  • From: "Sjoerd C. de Vries" <sjoerd.c.devries at gmail.com>
  • Date: Thu, 5 Mar 2009 04:58:47 -0500 (EST)
  • References: <200903031056.FAA02950@smc.vnet.net> <golqt7$qa1$1@smc.vnet.net>

Hi Bob,

You don't seem to have noticed that in the conjecture negative primes
were allowed as well. Therefore, the conjecture reads as
2n+1 = 2^i+p OR 2n+1 = 2^i - p, with p now defined as positive prime.
This is much harder to prove or disprove numerically because the
search space is infinite contrary to the case you examined.

Cheers -- Sjoerd

On Mar 4, 2:07 pm, DrMajorBob <btre... at austin.rr.com> wrote:
> The conjecture is false. It fails when the number tested is prime (but no=
t  
> the second of a twin prime pair). And it fails in other cases, too.
>
> Proof:
>
> Clear[test]
>   test[k_?OddQ] /; k >= 3 :=
>   Module[{n = 0},
>    Catch[While[2^n < k, PrimeQ[k - 2^n] && Throw@{2^n, k - 2^n, True}=
;
>      n++]; {k, False}]]
>
> These are the failures up to 1000:
>
> failures =
>   Cases[test /@ Range[3, 1000, 2], {k_, False} :> {k, PrimeQ@k}]
>
> {{127, True}, {149, True}, {251, True}, {331, True}, {337,
>    True}, {373, True}, {509, True}, {599, True}, {701, True}, {757,
>    True}, {809, True}, {877, True}, {905, False}, {907, True}, {959,
>    False}, {977, True}, {997, True}}
>
> Primes are marked with True, and non-primes with False, so the most  
> interesting of these is the first non-prime failure, 905.
>
> Here's an independent test for that one:
>
> 905 - 2^Range[0, Log[2, 905]]
> PrimeQ /@ %
>
> {904, 903, 901, 897, 889, 873, 841, 777, 649, 393}
>
> {False, False, False, False, False, False, False, False, False, False}
>
> Also, there are 2^k + 1 that fail:
>
> test /@ (1 + 2^Range[18])
>
> {{1, 2, True}, {2, 3, True}, {2, 7, True}, {4, 13, True}, {2, 31,
>    True}, {4, 61, True}, {2, 127, True}, {16, 241, True}, {4, 509,
>    True}, {4, 1021, True}, {32, 2017, True}, {4, 4093, True}, {2, 819=
1,
>     True}, {4, 16381, True}, {512, 32257, True}, {16, 65521, True}, {=
2,
>     131071, True}, {0, 262145, False}}
>
> The smallest of these is 2^18+1 == 262145.
>
> Bobby
>
> On Tue, 03 Mar 2009 04:56:33 -0600, 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 =1B$B!$=1B(BI 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?
>
> -- =
>
> DrMajor... at bigfoot.com



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