Thread: Minimum (Number Theory Game) View Single Post
2021-11-01, 23:11   #22
ThomasK

Aug 2021

11 Posts

Quote:
 Originally Posted by arbooker Interesting game. Maybe you know this already, but there is a simple proof that the Nullwertzahlen have density 0: any Nullwertzahl $$n$$ satisfies $$n\le2^{2^{\Omega(n)-1}}$$, so that $$\Omega(n)>\log\log(n)/\log(2)$$. As you pointed out in another post, $$\Omega(n)$$ is almost always close to $$\log\log{n}$$, so very few $$n$$ have so many prime factors. Following the proof of the Hardy-Ramanujan theorem, I think you can push this as far as a proof that $$W_0(x)=O(x/(\log{x})^\delta)$$ for some $$\delta>0$$, but I don't see how to get $$\delta=1$$ from just this.

Thanks very much. That's right; a randomly drawn natural number n has approximately ln ln n + O (1) prime factors, where the standard deviation is O ((ln ln n) ^ 0.5). The number of all natural numbers less than or equal to n that have exactly k prime factors is asyptotically n * (ln ln n) ^ (k-1) / ((ln n) * (k-1)!)

For k = 1, the prime number theorem results as a special case, which says that the number of prime numbers less than or equal to n is asymptotically about n / ln n.

For k = 2, all natural numbers are counted with exactly 2 prime factors, e.g. 4453 = 61 * 73 is a semi-prime number.

The number of all semi-prime numbers less than or equal to n is thus asymptotically n * ln ln n / ln n.
The number of all natural numbers less than or equal to n that have exactly 3 prime factors is thus asymptotically n * (ln ln n) ^ 2 / (ln n * 2)
The number of all natural numbers less than or equal to n that have exactly 4 prime factors is thus asymptotically n * (ln ln n) ^ 3 / (ln n * 6)
The number of all natural numbers less than or equal to n that have exactly 5 prime factors is thus asymptotically n * (ln ln n) ^ 4 / (ln n * 24)

etc.

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We are now considering extending the minimum total analysis, which was previously performed to 10 ^ 12, to 10 ^ 17. We do not yet believe that we will reach the smallest Fünfwertzahl (five-valued number) that we expect at around 10 ^ 18. Nevertheless, the minimum total analysis up to 10^17 would enable us to make a significantly improved extrapolation.

Anyone who has ideas on how to efficiently carry out a minimum total analysis that runs on 1000 processors at the same time, for example, can post his ideas in this thread.

How big should the basic block be?
How can the algorithm be efficiently parallelized?
Which programming language is best suited for the minimum total analysis up to 10^17?
Should the minimum total analysis be calculated up to 10^17 in a cloud?
How many and, if applicable, which prime numbers should be stored for the minimum total analysis up to 10^17?