Comments on: Deterministic way to find primes: discussion thread http://polymathprojects.org/2009/07/28/deterministic-way-to-find-primes-discussion-thread/ Massively collaborative mathematical projects Sat, 28 Apr 2012 14:00:10 +0000 hourly 1 http://wordpress.com/ By: Girish Varma http://polymathprojects.org/2009/07/28/deterministic-way-to-find-primes-discussion-thread/#comment-2430 Thu, 18 Nov 2010 16:35:02 +0000 http://polymathprojects.org/?p=70#comment-2430 I wanted to know if there is a solution to a slightly different but simple question:

Give a number n, how to find a prime larger than n in time bounded by a polynomial in \log n?

]]> By: Bhupinder Singh Anand http://polymathprojects.org/2009/07/28/deterministic-way-to-find-primes-discussion-thread/#comment-2369 Mon, 06 Sep 2010 06:05:52 +0000 http://polymathprojects.org/?p=70#comment-2369 “Find efficient deterministic algorithms for finding various types of “pseudoprimes” – numbers which obey some of the properties of being prime…”

The following define two deterministic prime generating algorithms which generate some curious composites, which are essentially a small (tiny?) subset of Hardy’s round numbers.

(Note that the focus is on generating the prime gaps, and not the primes themselves.)

TRIM NUMBERS

A number is Trim if, and only if, all its divisors are less than the Trim difference d_{n}, where:

(i) t_{1} = 2, and t_{n+1} = t_{n} + d_{n};

(ii) d_{1} = 1, and a(2, 1) = 1;

(iii) d_{n} is the smallest even integer that does not occur in the n’th sequence:

\{a(n, 1), \ldots, a(n, n-1)\};

(iv) j_{i} \geq 0 is selected so that:

0 \leq a(n+1, i) = ((a(n, i) - d(n) + j(i)*p_{i}) < p_{i}, where p_{i} is the i'th prime.

It follows that t_{n+1} is a prime unless all its prime divisors are less than d_{n}.

Question: What is the time taken to generate t_{n}?

COMPACT NUMBERS

A number is Compact if, and only if, all its divisors, except a maximum of one, are less than the Compact difference d_{n}, where:

(i) c_{1} = 2, and c_{n+1} = c_{n} + d_{n};

(ii) d_{1} = 1, and a(2, 1) = 1;

(iii) d_{n} is the smallest even integer that does not occur in the n’th sequence:

\{a(n, 1), \ldots, a(n, k)\};

(iv) j_{i} \geq 0 is selected so that, for all 0 < i \leq k:

0 \leq a(n+1, i) = (a(n, i) - d_{n} + j_{i}*p_{i}) < p_{i};

(v) k is selected so that:

p_{k}^{2} < c(n) \leq p_{k+1}^{2};

(vi) if c_{n} = p_{k+1}^{2}, then:

a(n, k+1) = 0.

It follows that c_{n+1} is either a prime, or a prime square, unless all, except a maximum of 1, prime divisors of the number are less than d_{n}.

What is the time taken to generate c_{n}?

The distribution of the Compact numbers suggests that the prime difference may be O(\pi(n)^{1/2}).

The algorithms can be seen in more detail at:

http://alixcomsi.com/A_Minimal_Prime.pdf

]]> By: Polymath again « What Is Research? http://polymathprojects.org/2009/07/28/deterministic-way-to-find-primes-discussion-thread/#comment-1043 Mon, 26 Oct 2009 22:43:08 +0000 http://polymathprojects.org/?p=70#comment-1043 [...] the meantime, Terence Tao started a polymath blog here, where he initiated four discussion threads (1, 2, 3 and 4) on deterministic ways to find primes (I’m not quite sure how that’s [...]

]]> By: Terence Tao http://polymathprojects.org/2009/07/28/deterministic-way-to-find-primes-discussion-thread/#comment-740 Mon, 31 Aug 2009 21:57:47 +0000 http://polymathprojects.org/?p=70#comment-740 Well, there are artificial counterexamples… \sum_{m \leq n} 2^m \mu(m), for instance, is computable in time poly(n) but has an error term which is exponential in n.

But yes, in general there seems to be a strong correlation between a quantity being easy to compute on one hand, and being easy to estimate on the other (Raphy Coifman is fond of pushing this particular philosophy).

]]> By: Mark Lewko http://polymathprojects.org/2009/07/28/deterministic-way-to-find-primes-discussion-thread/#comment-731 Mon, 31 Aug 2009 04:13:03 +0000 http://polymathprojects.org/?p=70#comment-731 Related to the comment above, I’d be interested to know if anyone is aware of any result of the following form: Give a non-trivial \Omega-bound for computing some non-trivial number-theoretic quantity.

]]> By: Mark Lewko http://polymathprojects.org/2009/07/28/deterministic-way-to-find-primes-discussion-thread/#comment-730 Mon, 31 Aug 2009 01:17:10 +0000 http://polymathprojects.org/?p=70#comment-730 Let me make a naive observation (based on four data points). During this project we have studied algorithms for computing the following number-theoretic quantities.

(1) \pi(n) = \sum_{p\leq n} 1

(2) D_{1}(n) = \sum_{x\leq n} \tau (x)

(3) D_{k}(n) = \sum_{x\leq n} \tau_{k}(x)

(4) Q(n) = \sum_{x\leq n} |\mu(x)|

In the following table, the first column indicates (ignoring log terms) how fast our best algorithm computes the above quantities, the second column indicates the best known error term on these quantities, and the third column indicates the conjectured error term on each of these quantities.

(1) n^{1/2} n n^{1/2}

(2) n^{1/3} n^{.314} n^{1/4}

(3) n^{1-1/k} n^{(k-1)/(k+2)} n^{(k-2)/2k}

(4) n^{1/3} n^{1/2} n^{.314}

In some cases (such as (1)) we can compute a quantity such that the order of the computation time is smaller than that of the best known error term. However, in none of these cases can we compute a quantity where the exponent on the computation time is better than the conjectured error term. This leads to the following rather vague question. Is anyone aware of any number theoretic quantity whose computation time is smaller than the conjectured error term?

]]> By: Terence Tao http://polymathprojects.org/2009/07/28/deterministic-way-to-find-primes-discussion-thread/#comment-707 Fri, 28 Aug 2009 05:13:06 +0000 http://polymathprojects.org/?p=70#comment-707 Dear Mark, one way to proceed is to first find a pair of consecutive residues b, b+1 mod W^2 that are not divisible by any square less than W^2 (this is possible by the Chinese remainder theorem). The density of square-frees in b mod W and in b+1 mod W are both close to 1, so there will be a lot of consecutive square-frees in this pair of progressions.

]]> By: Mark Lewko http://polymathprojects.org/2009/07/28/deterministic-way-to-find-primes-discussion-thread/#comment-686 Mon, 24 Aug 2009 23:12:36 +0000 http://polymathprojects.org/?p=70#comment-686 I have a question regarding the w-trick. It has been remarked in several places that we might be able to use the square-free gap results and the w-trick to find large pairs of consecutive square-free numbers. However, it isn’t clear to me how the details of this would work. The w-trick allows us to pass to an arithmetic progression with a density of square-free numbers arbitrary close to 1. Now, assuming that we can adapt the square-free gap results to arithmetic progressions (which seems reasonable to me), this would allow us to find consecutive square-free terms in that arithmetic progression, however this won’t produce a pair of genuinely consecutive square-free integers.

]]> By: Ernie Croot http://polymathprojects.org/2009/07/28/deterministic-way-to-find-primes-discussion-thread/#comment-651 Sat, 22 Aug 2009 17:31:52 +0000 http://polymathprojects.org/?p=70#comment-651 I typed in the paper to google, and found that Odlyzko has a link on his website at:

http://www.dtc.umn.edu/~odlyzko/doc/arch/analytic.comp.pdf

I looked up the section where it mentions the x^{2/3 + o(1)} algorithm, and apparently there is an analytic algorithm to compute \pi(x) in time x^{1/2+o(1)} using the zeta function (but not the explicit formula)! So it seems we were trying to solve a problem that has already been solved. Still, it would be good to have other solutions.

]]> By: Terence Tao http://polymathprojects.org/2009/07/28/deterministic-way-to-find-primes-discussion-thread/#comment-649 Sat, 22 Aug 2009 16:44:07 +0000 http://polymathprojects.org/?p=70#comment-649 According to this paper

Odlyzko, Andrew M.(1-BELL)
Analytic computations in number theory. (English summary) Mathematics of Computation 1943–1993: a half-century of computational mathematics (Vancouver, BC, 1993), 451–463,
http://www.ams.org/mathscinet-getitem?mr=1314883

there is an elementary algorithm to compute \pi(x) in x^{2/3+o(1)} time, though I don’t know whether this is a deterministic algorithm (I don’t have access to the paper). If so, that would solve this problem by binary search.

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