M7508981 has at least 12 prime factors(new factor # record)

[R.D. Silverman]

Originally Posted by alpertron

The previous values are for all Mersenne numbers with exponents less than 2³². I think it is better to filter to the range 1M to 10M (so we can perform the PRP test on the cofactor). In this way the statistics from http://www.mersenne.ca/manyfactors.p...n=4&fac_max=10 are:

10 prime factors known for 1 Mersenne number
9 prime factors known for 1 Mersenne number
8 prime factors known for 5 Mersenne numbers
7 prime factors known for 16 Mersenne numbers -> 0.003 % 
6 prime factors known for 114 Mersenne numbers -> 0.019 %
5 prime factors known for 682 Mersenne numbers -> 0.116 %
4 prime factors known for 4,337 Mersenne numbers -> 0.740 %

You are making a basic faulty assumption throughout your entire
discussion.

Consider the range of exponents that you are considering.
Let us say for the moment that it is [2, 50M]. Do you understand
that by Erdos-Kac, the number of expected factors varies WIDELY
over that range???


Furthermore, the data arises from factoring efforts that vary widely
across that range as well. You need to find ALL factors less than
N^alpha for a FIXED alpha as N varies from 2^2-1 to 2^~50M-1.

Over that range the SIZE of the composites changes from 1 digit to
~35 MILLION digits.

[alpertron]

Originally Posted by R.D. Silverman

You are making a basic faulty assumption throughout your entire
discussion.

Consider the range of exponents that you are considering.
Let us say for the moment that it is [2, 50M]. Do you understand
that by Erdos-Kac, the number of expected factors varies WIDELY
over that range???


Furthermore, the data arises from factoring efforts that vary widely
across that range as well. You need to find ALL factors less than
N^alpha for a FIXED alpha as N varies from 2^2-1 to 2^~50M-1.

It is impossible to find "ALL factors less than N^alpha for a FIXED alpha as N varies from 2^2-1 to 2^~50M-1" as you said because of the different sizes of the numbers we are considering. As that cannot be done (so the current theory does not help in this problem), I'm using the current statistics and the number of prime factors appears to follow a geometric distribution. That means that it is better to start with Mersenne numbers that already have a large number of prime factors, because the probability to find a new prime factor does not depend on the number of prime factors already found.

Over that range the SIZE of the composites changes from 1 digit to
~35 MILLION digits.

It is not 35 million, but 15 million.

[R.D. Silverman]

Originally Posted by alpertron

It is impossible to find "ALL factors less than N^alpha for a FIXED alpha as N varies from 2^2-1 to 2^~50M-1" as you said because of the different sizes of the numbers we are considering.

Of course it is impossible. But until you have such data, trying to
draw conclusions from your data is erroneous.

As that cannot be done (so the current theory does not help in this problem), I'm using the current statistics and the number of prime factors appears to follow a geometric distribution.

Which shows that your data is wrong/incomplete. re: Erdos-Kac

[alpertron]

What is the expected distribution of prime factors if we knew all prime factors below a certain constant bound B, instead of N^alpha?

[pdazzl]

Originally Posted by alpertron

I think it is better to filter to the range 1M to 10M (so we can perform the PRP test on the cofactor). In this way the statistics from http://www.mersenne.ca/manyfactors.p...n=4&fac_max=10 are:

I think that I am in agreement with you, the 1M to 10M/20M range may be the ideal breadth range to scan.

The crux of this problem comes back to current hardware limitations. Let's say 2^56 is a reasonable TF range to breadth scan exponents. A good question posed could be "Given a TF range of 2^56, what range of exponents will you find the highest density of exponents with discovered factors of 10 or more".

Another thing to consider is a prime factor can only divide one Mersenne number; thus as the exponents get bigger, the starting point of prime factors gets bigger (2p-1) and the density of primes gets smaller as primes get bigger one could infer that bigger exponents(over 1B) would need a higher TF range to reach the same density of discovered factors in an exponent range.

All the over 1B exponents with at least 8 known prime factors were discovered by me in the last couple weeks, there were no 1B+ exponents with at least 8 known prime factors just a few weeks ago. I have yet to bump an over 1B exponent up to 9 known prime factors.

[R.D. Silverman]

Originally Posted by alpertron

What is the expected distribution of prime factors if we knew all prime factors below a certain constant bound B, instead of N^alpha?

Define 'distribution'.

Do you want the counting function (Erdos-Kac) , or do you want the density function for factor sizes (Dickman's function)????

[wblipp]

Originally Posted by R.D. Silverman

This is correct. I do not know which would be more successful. I strongly suspect (based on some experience with similar things) that a breadth-first search will succeed with less expected work. I have not crunched through the numbers

I did crunch some numbers, but mostly learned that I need crunch a different set of numbers.

For various levels of "n" digits, I calculated the probability a number has 11 factors less than n digits. Then I calculated the ECM effort to clear that many candidates to n digits. "n" = 30 digits turned out to be the most efficient. This took much more time than running this one number to 45 digits (the median of finding the next factor).

But it's a stupid approach. The vast majority of the time is spent completing ECM to high levels for candidates that have few factors of small size. It's a waste of time because those numbers are very unlikely to have enough large factors to reach the total count of 11. I'm confident that some process of dropping those candidates with few factors at low levels will be much more effective.

A tip to others - don't forget that you cannot find 10^9 candidates with exponents near 10^7.

[alpertron]

Originally Posted by R.D. Silverman

Define 'distribution'.

Do you want the counting function (Erdos-Kac) , or do you want the density function for factor sizes (Dickman's function)????

I want to know the expected numbers in last table I presented to you:

10 prime factors known for 1 Mersenne number
9 prime factors known for 1 Mersenne number
8 prime factors known for 5 Mersenne numbers
7 prime factors known for 16 Mersenne numbers -> 0.003 % 
6 prime factors known for 114 Mersenne numbers -> 0.019 %
5 prime factors known for 682 Mersenne numbers -> 0.116 %
4 prime factors known for 4,337 Mersenne numbers -> 0.740 %

for the range of exponents 1M to 10M for the following factor bounds: 10²⁵, 10³⁰ and 10³⁵. You will need to replace the numbers 1, 1, 5, 16, 114, 682 and 4337 by the expected values.

[wblipp]

Originally Posted by alpertron

I want to know the expected numbers ... for the range of exponents 1M to 10M for the following factor bounds: 10²⁵, 10³⁰ and 10³⁵.

Here's what I did for exponent 10⁷ and factor bound 10³⁰.

We know there are not any factors below 10⁷. Beyond this we expect the number of factors to be typical. We know that the expected number of factors between 10^7 and 10^30 is ln(30/7) = 1.455. We expect the number of factors to be distributed Poisson. The probability of 11 or more in a Poisson random variable with this mean is 4.94 * 10^-8.

I'm not sure about the way I handled the small factor issue. For larger factors, I've previously looked at data that shows these estimates are reasonable in spite of the 2kn+1 issue. I guess I should look at a bunch of exponents near 10^7 and see how many factors show up between below10^10. My approach says it should be ln(10/7) = 0.357.

[pdazzl]

Originally Posted by wblipp

Here's what I did for exponent 10⁷ and factor bound 10³⁰.

We know there are not any factors below 10⁷. Beyond this we expect the number of factors to be typical. We know that the expected number of factors between 10^7 and 10^30 is ln(30/7) = 1.455. We expect the number of factors to be distributed Poisson. The probability of 11 or more in a Poisson random variable with this mean is 4.94 * 10^-8.

I'm not sure about the way I handled the small factor issue. For larger factors, I've previously looked at data that shows these estimates are reasonable in spite of the 2kn+1 issue. I guess I should look at a bunch of exponents near 10^7 and see how many factors show up between below10^10. My approach says it should be ln(10/7) = 0.357.

The 10th prime of the 7M # was 18 digits. So the probability that this scenario was possible would be based off ln(18/7)? Then probability multiplied by the number of primes between 1M and 10M?

[pdazzl]

I think that's the question we're looking for an anwer on...just how rare are these upper numbers that we've so far found in terms of probability. Is it a blue lobster, or an albino lobster? :)

http://www.thefeaturedcreature.com/2...-lobsters.html