98.00M to 98.05M Excluded Values

[Uncwilly]

Originally Posted by storm5510

I was tempted to drop it out of the list. If it will not hurt anything, I will. It can always be picked up later. Below is the value to "unreserve".

98000027,71,72

According to the Prime95 V25.9 help file exponents upto 96830000 are factored only to 71 bits. So you would be doing the last bit. Someone should do P-1 first. It won't hurt to drop it. You can take it out of your worktodo, then unreserve it or just let it sit and Primenet will take of it.

I post the following about factoring cut-off limits, final bit depths, end level of trial factoring, how far to search, so that others might find it by searching (thus all of the synonym terms):

Exponents up to	Trial factored to
29690000 	2^66
37800000 	2^67
47450000 	2^68
58520000 	2^69
75670000 	2^70
96830000 	2^71

[storm5510]

These are also being sent to the PrimeNet server

M98024831 no factor from 2^65 to 2^66
M98024837 no factor from 2^65 to 2^66
M98024881 no factor from 2^65 to 2^66
M98024887 no factor from 2^65 to 2^66
M98024897 no factor from 2^65 to 2^66
M98024957 no factor from 2^65 to 2^66
M98024957 no factor from 2^65 to 2^66
M98000059 no factor from 2^65 to 2^66
M98025013 no factor from 2^65 to 2^66
M98000069 no factor from 2^65 to 2^66
M98025119 no factor from 2^65 to 2^66

[Uncwilly]

Originally Posted by storm5510

These are also being sent to the PrimeNet server

You will feel a 'special thrill' when you see that you have found a factor.

Glad you have things rolling.

[storm5510]

Okay, you've got me confused now.

A factor is a value that evenly divides into another number. That would suggest to me that an exponent cannot be prime.

What am I missing?

[Uncwilly]

Originally Posted by storm5510

Okay, you've got me confused now.
...That would suggest to me that an exponent cannot be prime.

Finding a factor indicates that the exponent cannot be prime, true. However, by finding a factor saves a long LL test. By eliminating exponents via trial factoring and P-1, this saves a vast amount of CPU time that would otherwise be spent on LL test on exponents that aren't prime.

Here is how we can find these monsterously large primes so effciently:
First, we are working on a special form of number, that elimates us worrying about any number that is not expressible in this form. 2^x-1 This eliminates most numbers. (This step requires no effort)
Second, this class has a special property, it can only be prime if the exponent is prime. 2^{[COLOR="Blue"]p[/COLOR]}-1 This eliminates most of the remaining numbers. (This step requires effort that is so slight that it is of no consequence.)
Third, this class has a second special property, all factors of these numbers must fall into a pattern. 2kp+1 This eliminates many, many numbers that we might otherwise have to do try and divide the number by. We can eliminate a full 1/2 of the numbers that remain by checking for these factors. (This is the first step that requires real work, but the benefit is huge.)
Third and a half, there is a second type of test that can be done that can find more factors that are larger (P-1 testing), this can eliminate ~5-8% of those numbers that still remain. (This step requires more work, but yields good reward and data).
Lastly, these numbers can be tested by a 'tricky' test to check to see if they are prime, without having to try and divide the number by all primes below it's square root. (LL testing) This type of test makes it conceivable to test numbers that have 10 million digits. (This step requires the most work per number, but is the only step that can prove a number prime.)

Thus, finding a factor of a single number speeds this project on to finding the next prime.

edit: in the 100m digit range, we have already eliminater 56% of the numbers in the first group, 28600 of 51080

[storm5510]

It is really about saving time, and that's something I can go along with. It would be horrible to spend four or five month working on something that is not prime that could have been eliminated much sooner.

What does k represent in 2kp+1?

[10metreh]

Originally Posted by storm5510

It is really about saving time, and that's something I can go along with. It would be horrible to spend four or five month working on something that is not prime that could have been eliminated much sooner.

What does k represent in 2kp+1?

k is the variable. All factors of 2^p-1 are of the form 2kp+1, where k is any whole number and p is the exponent. Prime95 will use these numbers as possible factors and only trial divides by them, as others cannot be factors.

[cheesehead]

storm5510,

I think you may have been confused by Uncwilly's using a piece of forum jargon.

Originally Posted by storm5510

A factor is a value that evenly divides into another number. That would suggest to me that an exponent cannot be prime.

No, that is not a mathematical consequence.

Example: 2¹¹-1 has a factor (two of them actually), even though 11 is prime.

It is quite possible for a Mersenne number with a prime exponent to be composite -- in fact, the vast majority (99.99+%) are.

Perhaps you have confused that fact with the (different) statement that
a Mersenne number cannot be prime unless its exponent is prime.

As for confusion: it is a common jargon practice in this forum to use the unqualified word "exponent" to mean the Mersenne number having that exponent (p in 2^p-1). If p = 9876553, for instance, some folks here will refer to 2^9876553-1 by writing merely "9876553", as in "I'm testing 9876553" -- because all long-time members understand that shortcut. They know that really meant "I'm testing 2^9876553-1."

Originally Posted by Uncwilly

Finding a factor indicates that the exponent cannot be prime, true.

What you meant was that finding a factor of a Mersenne number with a certain exponent does not mean that the exponent cannot be prime. Right?

By eliminating exponents

... but what you really meant there is something like "By eliminating Mersenne numbers, usually designated by just their exponents, that were possible candidates for being primes ..." Right?

< snip > on LL test on exponents that aren't prime.

You meant something like "... on LL tests of Mersenne numbers that aren't prime (even though their exponents are prime)." Right?

[10metreh]

Originally Posted by cheesehead

As for confusion: it is a common jargon practice in this forum to use the unqualified word "exponent" to mean the Mersenne number having that exponent (p in 2^p-1). If p = 9876553, for instance, some folks here will refer to 2^9876553-1 by writing merely "9876553", as in "I'm testing 9876553" -- because all long-time members understand that shortcut. They know that really meant "I'm testing 2^9876553-1."

M9876553 has a factor: 34528429289

[cheesehead]

Originally Posted by 10metreh

M9876553 has a factor: 34528429289

... and that's what whoever tested it found.

[storm5510]

2¹¹-1 is 2047. It has a factor? I believe George's phrase goes, "If p is prime, then 2^p-1 is prime." Well, 11 is not prime. 2047 is odd, but that doesn't mean much. What is the factor?