20140303, 23:06  #1 
Oct 2013
7 Posts 
Binary pattern in p for Mersenne prime 2p1
The p from 2^{p}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 there a pattern how the ones and zeros are following each other in p? Binary format for p where 2^{p}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 You can verify the binary formats on page http://www.mathsisfun.com/binarydec...converter.html Last fiddled with by TrdH on 20140303 at 23:08 Reason: Moved smiley... 
20140303, 23:26  #2  
Nov 2003
1110100100100_{2} Posts 
Quote:
of statistics. I will offer a hint: Look up "runs test" for a Bernoulli process. 

20140303, 23:32  #3 
May 2013
East. Always East.
11×157 Posts 
[TROLL]No, Mr. Silverman. There is a clear pattern. Each 0 is followed by either a 1 or a 0.[/TROLL]

20140304, 00:22  #4 
Jul 2003
So Cal
2^{2}·11·47 Posts 
They all both begin and end with a 1!

20140304, 00:41  #5 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
7×11×79 Posts 
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.

20140304, 00:59  #6 
Aug 2006
2×29×103 Posts 
32582657 has 25 bits. The average maximal run of 1bits in 25bit primes is about 4.14, so it's not particularly noteworthy or unusual that 32582657 has a run of 5 1bits. About a third (109881/328606) of the primes of that size do.

20140304, 01:06  #7 
"Mike"
Aug 2002
17460_{8} Posts 
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. 
20140304, 01:14  #8 
"Mike"
Aug 2002
1111100110000_{2} Posts 
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. 
20140304, 02:10  #9 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}·2,333 Posts 
The most important pattern in Mersenne primes is that in binary they are all ones!
But wait, there's more! 
20140304, 04:03  #10 
May 2013
East. Always East.
3277_{8} Posts 
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 requote 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 20140304 at 04:09 
20140304, 04:17  #11 
May 2013
East. Always East.
11×157 Posts 
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 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^{(x1)}. In a string of length y bits, there are yx+1 starting points for a string like that. The odds of having a matching piece of string are therefore (yx+1)*2^{(x1)} 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 20140304 at 04:32 
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