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 2002-11-04, 05:10 #1 TTn   2·37·131 Posts Mersenne's on Ulams spiral I was messing with the prime number spiral, and decided to try odd numbers instead. To my amazment odd mersenne numbers, fall onto one strip. In the form of the polynomial 8(n^2) -1. and n is power of two. The numbers that interest me here, are those of the above form that are not mersenne, but are prime. The study of their density, could be used for finding mersenne primes? :D
2002-11-04, 12:44   #2
toferc

Aug 2002

368 Posts
Re: Mersenne's on Ulams spiral

Quote:
 Originally Posted by TTn ... In the form of the polynomial 8(n^2) -1. and n is power of two.
This form can be rewritten as 2^k - 1, where k is odd and greater than one, thus these numbers form a subset of the Mersenne numbers. It is known that 2^k - 1 is prime only if k is prime, therefore all Mersenne primes other than 3 are in this sequence, and they are the only primes in this sequence.

 2002-11-04, 21:56 #3 TTn   553010 Posts Obviously you weren't listening carefully. 2^k -1 is a specific form. The densitiy of primes, in the subset form 8(n^2) -1 is the real question, which in turn, effects the density of primes 2^k -1.
 2002-11-04, 22:14 #4 TTn   252010 Posts Clarification, ALL values N. The densitiy of primes, in the subset form 8(n^2) -1 Also the densitity of primes in form 8(n^2) +1 /3 so the probability that they may both prime can be analyzed, in respect to the new mersenne conjecture.
 2002-11-05, 02:22 #5 Deamiter     Sep 2002 32×13 Posts it seems to me that 8(n^2)-1 is a subset of 2^k-1 as long as 'n' is a power of two like you said. That's because 8=2^3 so your form is really (2^3)*(n^2)-1, but since n is always a power of two, you can always simplify the number into the form 2^k-1... I hope you can see why this is... as any (power of two)^2 can always be written in the form 2^n where n will always be even. so your numbers are really in the form 2^(n+3)-1... and now I'm getting tired, so if that made little sense, wait for one of the GIMPS gurus to respond with a better explanation.
 2002-11-05, 10:54 #6 TTn   29×239 Posts "seems to me that 8(n^2)-1 is a subset of 2^k-1 as long as 'n' is a power of two like you said. " Stop ! then I go on to say, the interesting numbers are in the above form, BUT are NOT mersenne. This means I am speaking of all other n, not producing a mersenne number(odd specifically), but such that produce a prime number. The density could be manipulated, to analyze that of mersenne primes. If we were talking about the same thing, then there would be primes like, 2^k -1 = 199, where k is not an integer. I hope this clears it up.
 2002-11-05, 17:38 #7 ewmayer ∂2ω=0     Sep 2002 República de California 5×2,351 Posts What you've observed is simply a "rich vein" of primes of a certain form. The folks searching for huge Proth primes (p = k*2^n + 1) also look for such veins (in their case, k's that yield a high percentage of primes) to maximize their chances of finding one. In your case, it shouldn't be surprising that there are many primes of the form N = 8*n^2 - 1. If we write this as 2^3*n^2 - 1, we see that unless n = 2^j and 2*j+3 is composite (i.e. a composite-exponent Mersenne), the fact that the power on the 2 and the n are relatively prime means we can't write N as a^k - b^k (in your case b = 1), which rules out any easy algebraic factors. That significantly boosts the odds of N being prime. It's also quite likely (I'll leave you to investigate this) that N of this form, if they are composite, must have their factors of a certain algebraic form. That would again increase the prime density, by eliminating most random odd numbers as candidate factors. Yes, the Mersennes are a special case of these numbers, but it seems doubtful that fact will help us find more Mersennes - those are a special case unto themselves, we know a lot about their properties and even have a fast deterministic primality test for them. We also know they occur (AFAWK) for random prime exponents of 2, and that their density thins out fairly predictably as they get bigger. FYI, here are all the n <= 10^4 for which 2^3*n^2 - 1 is prime. You see the general thinning-out trend, with a few hiccups (e.g. the n = 5000-6000 and 8000-9000 columns are longer than the ones that precede them), but one sees such behavior for most any reasonably dense sequence of primes one wishes to investigate: [code:1] 1 1003 2004 3013 4006 5012 6001 7003 8008 9016 2 1004 2012 3014 4008 5014 6002 7007 8012 9018 3 1008 2020 3015 4011 5015 6004 7018 8013 9027 4 1011 2021 3017 4037 5017 6013 7023 8018 9033 5 1018 2023 3022 4046 5033 6017 7037 8029 9040 9 1026 2026 3027 4048 5035 6020 7038 8032 9041 11 1032 2034 3031 4057 5036 6024 7049 8039 9042 12 1033 2035 3035 4058 5052 6027 7051 8040 9056 14 1034 2051 3038 4067 5058 6029 7056 8046 9065 17 1036 2053 3049 4071 5065 6034 7061 8052 9074 18 1038 2068 3057 4074 5068 6037 7063 8060 9077 19 1039 2072 3061 4076 5075 6044 7067 8075 9082 21 1041 2083 3062 4077 5085 6052 7072 8078 9086 23 1045 2090 3068 4081 5092 6055 7094 8081 9089 25 1048 2095 3077 4084 5098 6060 7098 8087 9090 26 1050 2103 3080 4088 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your comment about N = 2^k - 1 for non-integer k: if I allow k to be irrational, I can write any positive integer I want this way. So what?
 2002-11-06, 02:41 #8 TTn   22·3·617 Posts Thanks for the list! I doubt, the doubt. The comment 2^k-1 was to verify that we were not on the same page.

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