The Prime Web |
Another interesting result from the analyses above has to do with the mersenne primes. The first mersenne prime numbers are:
no | p-exp | prime | ||
1. | 2 | 3 | ||
2. | 3 | 7 | ||
3. | 5 | 31 | ||
4. | 7 | 127 | ||
5. | 13 | 8191 | ||
6. | 17 | 131071 | ||
7. | 19 | 524287 | ||
8. | 31 | 2147483647 | ||
9. | 61 | 2305843009213693951 |
As you can see the mersenne primes occur neatly in the category of “faked” composite numbers - that is when the composite numbers are treated as if they were primes and described as rows with 1´s. In the prime web these numbers “do not exist” as graphic number patterns, but of course you may construct any pattern in the binary system.
It seems like the only primes you get if you convert binary rows of 1´s to decimal numbers are the mersennes primes. Another interesting thing is that the mersenne primes occur later on in the number list as the largest and last factor in the number which is twice as big as the original number minus 1.
If you for instance look at the mersenne prime for number 14 (that is 2^{13}-1) it gives the mersenne prime 8191. If you then go to (2·14-1=27) you get
67108863 = 3 · 2731 · 8191.This seems to be the case for all mersenne primes. Here is the list for every binary row of 1´s up to 63 converted to the decimal system and then factorized. And in the end we focus on some higher mersenne primes to show that the theory seems to be valid. The mersenne primes are marked with turquoise and their double with yellow.
As you can see the mersenne primes are found together with the even number just above the prime exponent we are used to connect with the number. This comes from the fact that the row of 1´s contains one less 1 than the exponent (for instance number 13 contains 12 filled squares, that is 1´s), which derives from the construction of the number web where each number is marked with a last empty square.
Click here to see the list.
This means that you can conclude if a prime is a mersenne prime or not if you go to 2^{2p} – 1 (where p is the prime exponent we are used to, but actually refers to n-1) and find that the last factor is the same as the prime number you investigate.
What is also interesting here is that the expression 2p actually refers to
((2(p+1) -1)-1). As you can see from the list the mersenne prime first occur at a row of an even number of 1´s (binary system), for instance 8191 at the number 14. To get to the second place in the list you have take (2·14-1) and then you always get an odd number. Sometimes this odd number is a prime. The first times this happens is for the mersenne prime numbers 7, 31 and 618970019642690137449562111 (see the list above). These mersenne primes are therefore some very special numbers.
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© Gunnar Appelqvist 2009
Latest update 2009-01-29