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Old 2014-03-03, 23:06   #1
TrdH
 
Oct 2013

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Question Binary pattern in p for Mersenne prime 2p-1

The p from 2p-1 when it's a Mersenne prime
has short chains of ones in binary format for the 48 Mersenne primes.

Here are the Mersenne primes with at least 5 ones in a chain for p in binary format from the 48 Mersenne primes found - using list from http://primes.utm.edu/mersenne/index.html:

Code:
              11111 - 31 - M8 - 100 - 5 most
            1111111 - 127 - M12 - 1100 - 7 most
         1001011111 - 607 - M14 - 1110 - 5 most
        10011111111 - 1279 - M15 - 1111 - 8 most
         11010111110100111 - 110503 - M29 - 11101 - 5 most
101010101010111111101 - 1398269 - M35 - 100011 - 7 most
     110011010111110100100101 - 13466917 - M39 - 100111 - 5 most
 1010000000101111110101011 - 20996011 - M40 - 101000 - 6 most
       1110011111110011110011001 - 30402457 - M43 - 101011 - 7 most
              1111100010010110000000001 - 32582657 - M44? - 101100 - 5 most
Is it less probable that that there are other Mersenne primes (2p-1) with 8 or longer chains of ones for p in binary format?


Is there a pattern how the ones and zeros are following each other in p?

Binary format for p where 2p-1 is Mersenne prime (hopped over the Mersenne primes mentioned above):

Code:
11011100110100000111101001 - 57885161 - M48? - 110000?
10100100011101100010100001 - 43112609 - M47? - 101111?
10100010101011000101011001 - 42643801 - M46? - 101110?
10001101101111011100111011 - 37156667 - M45? - 101101?
 1100011000011000110010111 - 25964951 - M42 - 101010
 1011011101100010011100111 - 24036583 - M41 - 101001
   11010100110010010110001 -  6972593 - M38 - 100110
    1011100001101001000001 -  3021377 - M37 - 100101
    1011010110100111011101 -  2976221 - M36 - 100100
     100110011000100111011 -  1257787 - M34 - 100010
      11010001110100101001 -   859433 - M33 - 100001
      10111000110001100111 -   756839 - M32 - 100000
        110100110000011011 -   216091 - M31 - 11111
        100000001111010001 -   132049 - M30 - 11110
         10101000011100011 -    86243 - M28 - 11100
          1010110111010001 -    44497 - M27 - 11011
           101101010101001 -    23209 - M26 - 11010
           101010011000101 -    21701 - M25 - 11001
           100110111100001 -    19937 - M24 - 11000
            10101111001101 -    11213 - M23 - 10111
            10011011010101 -     9941 - M22 - 10110
            10010111011001 -     9689 - M21 - 10101
             1000101000111 -     4423 - M20 - 10100
             1000010011101 -     4253 - M19 - 10011
              110010010001 -     3217 - M18 - 10010
              100011101001 -     2281 - M17 - 10001
              100010011011 -     2203 - M16 - 10000
                1000001001 -      521 - M13 - 1101
                   1101011 -      107 - M11 - 1011
                   1011001 -       89 - M10 - 1010
                    111101 -       61 - M9 - 1001
                     10011 -       19 - M7 - 111
                     10001 -       17 - M6 - 110
                      1101 -       13 - M5 - 101
                       111 -        7 - M4 - 100
                       101 -        5 - M3 - 11
                        11 -        3 - M2 - 10
                        10 -        2 - M1 - 1
The last column is the place in the Mersenne prime list in binary format - is there a relation there too?


You can verify the binary formats on page http://www.mathsisfun.com/binary-dec...converter.html

Last fiddled with by TrdH on 2014-03-03 at 23:08 Reason: Moved smiley...
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Old 2014-03-03, 23:26   #2
R.D. Silverman
 
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Quote:
Originally Posted by TrdH View Post
The p from 2p-1 when it's a Mersenne prime
has short chains of ones in binary format for the 48 Mersenne primes.

Here are the Mersenne primes with at least 5 ones in a chain for p in binary format from the 48 Mersenne primes found - using list from http://primes.utm.edu/mersenne/index.html:

Code:
              11111 - 31 - M8 - 100 - 5 most
            1111111 - 127 - M12 - 1100 - 7 most
         1001011111 - 607 - M14 - 1110 - 5 most
        10011111111 - 1279 - M15 - 1111 - 8 most
         11010111110100111 - 110503 - M29 - 11101 - 5 most
101010101010111111101 - 1398269 - M35 - 100011 - 7 most
     110011010111110100100101 - 13466917 - M39 - 100111 - 5 most
 1010000000101111110101011 - 20996011 - M40 - 101000 - 6 most
       1110011111110011110011001 - 30402457 - M43 - 101011 - 7 most
              1111100010010110000000001 - 32582657 - M44? - 101100 - 5 most
Is it less probable that that there are other Mersenne primes (2p-1) with 8 or longer chains of ones for p in binary format?


Is there a pattern how the ones and zeros are following each other in p?

Binary format for p where 2p-1 is Mersenne prime (hopped over the Mersenne primes mentioned above):

Code:
11011100110100000111101001 - 57885161 - M48? - 110000?
10100100011101100010100001 - 43112609 - M47? - 101111?
10100010101011000101011001 - 42643801 - M46? - 101110?
10001101101111011100111011 - 37156667 - M45? - 101101?
 1100011000011000110010111 - 25964951 - M42 - 101010
 1011011101100010011100111 - 24036583 - M41 - 101001
   11010100110010010110001 -  6972593 - M38 - 100110
    1011100001101001000001 -  3021377 - M37 - 100101
    1011010110100111011101 -  2976221 - M36 - 100100
     100110011000100111011 -  1257787 - M34 - 100010
      11010001110100101001 -   859433 - M33 - 100001
      10111000110001100111 -   756839 - M32 - 100000
        110100110000011011 -   216091 - M31 - 11111
        100000001111010001 -   132049 - M30 - 11110
         10101000011100011 -    86243 - M28 - 11100
          1010110111010001 -    44497 - M27 - 11011
           101101010101001 -    23209 - M26 - 11010
           101010011000101 -    21701 - M25 - 11001
           100110111100001 -    19937 - M24 - 11000
            10101111001101 -    11213 - M23 - 10111
            10011011010101 -     9941 - M22 - 10110
            10010111011001 -     9689 - M21 - 10101
             1000101000111 -     4423 - M20 - 10100
             1000010011101 -     4253 - M19 - 10011
              110010010001 -     3217 - M18 - 10010
              100011101001 -     2281 - M17 - 10001
              100010011011 -     2203 - M16 - 10000
                1000001001 -      521 - M13 - 1101
                   1101011 -      107 - M11 - 1011
                   1011001 -       89 - M10 - 1010
                    111101 -       61 - M9 - 1001
                     10011 -       19 - M7 - 111
                     10001 -       17 - M6 - 110
                      1101 -       13 - M5 - 101
                       111 -        7 - M4 - 100
                       101 -        5 - M3 - 11
                        11 -        3 - M2 - 10
                        10 -        2 - M1 - 1
The last column is the place in the Mersenne prime list in binary format - is there a relation there too?


You can verify the binary formats on page http://www.mathsisfun.com/binary-dec...converter.html
Typical numerology nonsense from someone with no understanding
of statistics. I will offer a hint: Look up "runs test" for a Bernoulli
process.
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Old 2014-03-03, 23:32   #3
TheMawn
 
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[TROLL]No, Mr. Silverman. There is a clear pattern. Each 0 is followed by either a 1 or a 0.[/TROLL]
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Old 2014-03-04, 00:22   #4
frmky
 
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They all both begin and end with a 1!
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Old 2014-03-04, 00:41   #5
retina
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Quote:
Originally Posted by TrdH View Post
Is there a pattern how the ones and zeros are following each other in p?
I think you are on to something here. It is amazing that no one else has ever thought about this before. I hope that you can continue your important work in this area and can come up with the answer. You can ignore all the other posters here, they are just jealous because they didn't think of it first. So please keep posting all of your amazing work here. We will read it and treat it will all the care and attention it deserves.
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Old 2014-03-04, 00:59   #6
CRGreathouse
 
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32582657 has 25 bits. The average maximal run of 1-bits in 25-bit primes is about 4.14, so it's not particularly noteworthy or unusual that 32582657 has a run of 5 1-bits. About a third (109881/328606) of the primes of that size do.
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Old 2014-03-04, 01:06   #7
Xyzzy
 
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It is perhaps easy for real smart people to tease other people who post topics like this.

We admit that we search for patterns all of the time. We do not (yet) have the sophisticated knowledge to understand why we are barking up the wrong tree, but as the years have gone by we are slowly getting a feel of what is possible and what is not.

If this forum works as intended, the overall goal might be to show the OP how this approach is futile, or maybe link the OP to documentation to that effect.

It is true that many of the people whose posts end up in this particular subforum do not want help, but that is not always the case.

Personally, we learn a little from every post, especially when things are explained a bit.

There are probably many people with questions such as the one posed this thread who are afraid to ask them. We know that we hesitate to ask questions due to our poor ability to state things mathematically.

We are beginning to ramble a bit here so we will close our thoughts with the suggestion that we all might try to remember that the OP is a real person somewhere and deserves, at least initially, some respect.
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Old 2014-03-04, 01:14   #8
Xyzzy
 
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To be fair, Mister Silverman did suggest reading material. We got as far as:

http://www.itl.nist.gov/div898/handb...on3/eda35d.htm

Unfortunately, the contents of that page are beyond our level of mathematical understanding.

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Old 2014-03-04, 02:10   #9
Batalov
 
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The most important pattern in Mersenne primes is that in binary they are all ones!

But wait, there's more! If you call within the next 15 minutes... All other primes have at least one zero in their binary representation.
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Old 2014-03-04, 04:03   #10
TheMawn
 
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Well, to be fair, Retina is right that it may be worth a shot. However, I must admit my disdain for the "hey guys here's a bunch of numbers is there a pattern?" approach.

Perhaps more than four minutes of thought could have gone into the whole thing before bringing it up. I just feel like it's a wee bit demanding on US as readers to interpret someone else's numbers that haven't got much backing.

On the other hand, I am going to re-quote the numbers but this time I will include the skipped ones because it's hard to see a pattern with missing values.

Code:
11011100110100000111101001 - 57885161 - M48?- 110000?
10100100011101100010100001 - 43112609 - M47?- 101111?
10100010101011000101011001 - 42643801 - M46?- 101110?
10001101101111011100111011 - 37156667 - M45?- 101101?
 1111100010010110000000001 - 32582657 - M44?- 101100
 1110011111110011110011001 - 30402457 - M43 - 101011  
 1100011000011000110010111 - 25964951 - M42 - 101010
 1011011101100010011100111 - 24036583 - M41 - 101001
 1010000000101111110101011 - 20996011 - M40 - 101000
  110011010111110100100101 - 13466917 - M39 - 100111
   11010100110010010110001 -  6972593 - M38 - 100110
    1011100001101001000001 -  3021377 - M37 - 100101
    1011010110100111011101 -  2976221 - M36 - 100100
     101010101010111111101 -  1398269 - M35 - 100011
     100110011000100111011 -  1257787 - M34 - 100010
      11010001110100101001 -   859433 - M33 - 100001
      10111000110001100111 -   756839 - M32 - 100000
        110100110000011011 -   216091 - M31 - 11111
        100000001111010001 -   132049 - M30 - 11110
         11010111110100111 -   110503 - M29 - 11101
         10101000011100011 -    86243 - M28 - 11100
          1010110111010001 -    44497 - M27 - 11011
           101101010101001 -    23209 - M26 - 11010
           101010011000101 -    21701 - M25 - 11001
           100110111100001 -    19937 - M24 - 11000
            10101111001101 -    11213 - M23 - 10111
            10011011010101 -     9941 - M22 - 10110
            10010111011001 -     9689 - M21 - 10101
             1000101000111 -     4423 - M20 - 10100
             1000010011101 -     4253 - M19 - 10011
              110010010001 -     3217 - M18 - 10010
              100011101001 -     2281 - M17 - 10001
              100010011011 -     2203 - M16 - 10000
               10011111111 -     1279 - M15 - 1111
                1001011111 -      607 - M14 - 1110
                1000001001 -      521 - M13 - 1101
                   1111111 -      127 - M12 - 1100
                   1101011 -      107 - M11 - 1011
                   1011001 -       89 - M10 - 1010
                    111101 -       61 - M9  - 1001
                     11111 -       31 - M8  - 100
                     10011 -       19 - M7  - 111
                     10001 -       17 - M6  - 110
                      1101 -       13 - M5  - 101
                       111 -        7 - M4  - 100
                       101 -        5 - M3  - 11
                        11 -        3 - M2  - 10
                        10 -        2 - M1  - 1

Last fiddled with by TheMawn on 2014-03-04 at 04:09
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Old 2014-03-04, 04:17   #11
TheMawn
 
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Code:
11011100110100000111101001 - 57885161 - M48?- 110000?
10100100011101100010100001 - 43112609 - M47?- 101111?
10100010101011000101011001 - 42643801 - M46?- 101110?
10001101101111011100111011 - 37156667 - M45?- 101101?
 1111100010010110000000001 - 32582657 - M44?- 101100
 1110011111110011110011001 - 30402457 - M43 - 101011  
 1100011000011000110010111 - 25964951 - M42 - 101010
 1011011101100010011100111 - 24036583 - M41 - 101001
 1010000000101111110101011 - 20996011 - M40 - 101000
  110011010111110100100101 - 13466917 - M39 - 100111
   11010100110010010110001 -  6972593 - M38 - 100110
    1011100001101001000001 -  3021377 - M37 - 100101
    1011010110100111011101 -  2976221 - M36 - 100100
     101010101010111111101 -  1398269 - M35 - 100011
     100110011000100111011 -  1257787 - M34 - 100010
      11010001110100101001 -   859433 - M33 - 100001
      10111000110001100111 -   756839 - M32 - 100000
        110100110000011011 -   216091 - M31 - 11111
        100000001111010001 -   132049 - M30 - 11110
         11010111110100111 -   110503 - M29 - 11101
         10101000011100011 -    86243 - M28 - 11100
          1010110111010001 -    44497 - M27 - 11011
           101101010101001 -    23209 - M26 - 11010
           101010011000101 -    21701 - M25 - 11001
           100110111100001 -    19937 - M24 - 11000
            10101111001101 -    11213 - M23 - 10111
            10011011010101 -     9941 - M22 - 10110
            10010111011001 -     9689 - M21 - 10101
             1000101000111 -     4423 - M20 - 10100
             1000010011101 -     4253 - M19 - 10011
              110010010001 -     3217 - M18 - 10010
              100011101001 -     2281 - M17 - 10001
              100010011011 -     2203 - M16 - 10000
               10011111111 -     1279 - M15 - 1111
                1001011111 -      607 - M14 - 1110
                1000001001 -      521 - M13 - 1101
                   1111111 -      127 - M12 - 1100
                   1101011 -      107 - M11 - 1011
                   1011001 -       89 - M10 - 1010
                    111101 -       61 - M9  - 1001
                     11111 -       31 - M8  - 100
                     10011 -       19 - M7  - 111
                     10001 -       17 - M6  - 110
                      1101 -       13 - M5  - 101
                       111 -        7 - M4  - 100
                       101 -        5 - M3  - 11
                        11 -        3 - M2  - 10
                        10 -        2 - M1  - 1
I have to actually post this before I can try to read it. Maybe this was just one big exercise in eyestrain and nothing else.


Let's calculate the odds of finding a matching string like that.

The odds of a string of length x bits being matching a given string of x bits is 2-(x-1).

In a string of length y bits, there are y-x+1 starting points for a string like that.

The odds of having a matching piece of string are therefore (y-x+1)*2-(x-1) if my math is right.

Let's try M24: The string is 5 bits long: 11000. P is 15 bits: 100110111100001.

There are 11 starting points available and the odds of that string appearing inside are 1 in 16.

If the odd are 11 in 16, then I find absolutely no statistical significance to the matching bits of string. And there we go.


EDIT: This was only a test to see if there was anything odd. As it happens, there isn't. Even if there was, a lot of "patterns" fail to follow for the next element, and I see no way of "predicting" the matching strings.

This kind of also was a test just for my own purposes which I figured I might as well put here [TROLL] so someone else can pick up my valuable and important work where I stopped but give me credit for giving them the idea if something comes of it [/TROLL]

Last fiddled with by TheMawn on 2014-03-04 at 04:32
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