mersenneforum.org Binary pattern in p for Mersenne prime 2p-1
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 2014-03-03, 23:06 #1 TrdH   Oct 2013 7 Posts 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...
2014-03-03, 23:26   #2
R.D. Silverman

Nov 2003

1D2416 Posts

Quote:
 Originally Posted by TrdH 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.

 2014-03-03, 23:32 #3 TheMawn     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]
 2014-03-04, 00:22 #4 frmky     Jul 2003 So Cal 3×17×41 Posts They all both begin and end with a 1!
2014-03-04, 00:41   #5
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

22×5×307 Posts

Quote:
 Originally Posted by TrdH 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.

 2014-03-04, 00:59 #6 CRGreathouse     Aug 2006 32×5×7×19 Posts 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.
 2014-03-04, 01:06 #7 Xyzzy     "Mike" Aug 2002 11111111000002 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.
 2014-03-04, 01:14 #8 Xyzzy     "Mike" Aug 2002 11111111000002 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.
 2014-03-04, 02:10 #9 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 942310 Posts 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.
 2014-03-04, 04:03 #10 TheMawn     May 2013 East. Always East. 11×157 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 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
 2014-03-04, 04:17 #11 TheMawn     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 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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