My Math Forum Conjecture about primes of the form 2^k-1

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 October 16th, 2015, 05:45 AM #11 Banned Camp   Joined: Dec 2013 Posts: 1,117 Thanks: 41 Building a NON TRIVIAL and INFINITE sequence of composites numbers where no element is divided by any p < B (B is bound equal for example to 10.000 (I mean the first prime < 10.000)) is interesting. It will be a big discovery. Last edited by mobel; October 16th, 2015 at 06:12 AM.
 October 16th, 2015, 06:12 AM #12 Member   Joined: Jun 2015 From: Ohio Posts: 99 Thanks: 19 Here's a conjecture I thought of that definitely seems interesting. It's held for the few random values I've checked... Of course I have yet to check any large values (I'll do that later). If $\displaystyle 2^k - 2^j - 1$ is composite for all j from j = 1 to k - 1 then $\displaystyle 2^k - 1$ is a Mersenne Prime. It's not an if and only if because not all Mersenne primes have this property, but what if having this property guarantees it is a Mersenne prime?
October 16th, 2015, 06:14 AM   #13
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Quote:
 Originally Posted by numberguru1 Here's a conjecture I thought of that definitely seems interesting. It's held for the few random values I've checked... Of course I have yet to check any large values (I'll do that later). If $\displaystyle 2^k - 2^j - 1$ is composite for all j from j = 1 to k - 1 then $\displaystyle 2^k - 1$ is a Mersenne Prime. It's not an if and only if because not all Mersenne primes have this property, but what if having this property guarantees it is a Mersenne prime?
k>5

 October 16th, 2015, 06:17 AM #14 Member   Joined: Jun 2015 From: Ohio Posts: 99 Thanks: 19 ? k>5 isn't a counterexample... k = 6 works
October 16th, 2015, 06:30 AM   #15
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 Originally Posted by numberguru1 ? k>5 isn't a counterexample... k = 6 works
Sorry. I did a mistake.
When k=3

2^3-1=7

7-2=5
7-4=3
Both are prime not composite

2^3-1 is a Mersenne prime so you have to bound k

2^5-1 is also a Mersenne prime

Anyway you have to bound k

Last edited by mobel; October 16th, 2015 at 06:33 AM.

October 16th, 2015, 06:45 AM   #16
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 Originally Posted by mobel Primes are not random. If you start by this obvious observation then it surely exist a thick range containing at least one prime.
Yes, that observation is obvious. But you misunderstood my point -- it's thinness, not thickness, that made the Heath-Brown proof interesting. (Sorry, I reread my post and it was confusing. Heath-Brown's sequence is somewhat thin -- less than x^(2/3) numbers up to x. You're asking about much thinner sequences, and so that would be far harder in general.)

Last edited by CRGreathouse; October 16th, 2015 at 06:53 AM.

October 16th, 2015, 06:46 AM   #17
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Quote:
 Originally Posted by mobel Building a NON TRIVIAL and INFINITE sequence of composites numbers where no element is divided by any p < B (B is bound equal for example to 10.000 (I mean the first prime < 10.000)) is interesting. It will be a big discovery.
I'd be happy to try -- I have some ideas in mind -- but I'll need a definition of "NON TRIVIAL" first.

October 16th, 2015, 06:51 AM   #18
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Quote:
 Originally Posted by numberguru1 Here's a conjecture I thought of that definitely seems interesting. It's held for the few random values I've checked... Of course I have yet to check any large values (I'll do that later). If $\displaystyle 2^k - 2^j - 1$ is composite for all j from j = 1 to k - 1 then $\displaystyle 2^k - 1$ is a Mersenne Prime. It's not an if and only if because not all Mersenne primes have this property, but what if having this property guarantees it is a Mersenne prime?
It fails at 1, 15, 23, 27, 37, 39, 43, 55, 58, 63, 71, 79, 82, 91, 95, 111, 123, 133, 135, 139, 143, 148, 151, 159, 167, 169, 172, 173, 175, 179, 183, 191, 195, 199, 207, 211, 223, 239, 255, 286, 295, 313, 316, 319, 335, 337, 351, 367, 373, 383, 406, 415, 417, 433, 435, 447, 455, 461, 463, 479, 483, 487, 493, 495, 497, 505, 511, 517, 523, 527, 543, 551, 559, 571, 575, 578, 583, 587, 591, 599, 603, 615, 623, 635, 639, 643, 651, 655, 671, 673, 679, 682, 687, 703, 711, 715, 735, 739, 742, 745, 751, 767, 771, 783, 787, 799, 803, 805, 807, 811, 815, 823, 827, 831, 839, 847, 863, 867, 871, 887, 895, 911, 915, 923, 927, 943, 959, 963, 967, 975, 991, 999, ...

October 16th, 2015, 06:58 AM   #19
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 Originally Posted by CRGreathouse I'd be happy to try -- I have some ideas in mind -- but I'll need a definition of "NON TRIVIAL" first.
(k*10.000!)+1 is a trivial one

October 16th, 2015, 07:22 AM   #20
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 Originally Posted by mobel (k*10.000!)+1 is a trivial one
I was hoping for a definition rather than an example. I don't want to spend time on the problem until I know what it is.

 Tags 2k1, conjecture, form, primes

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