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Message boards : Sierpinski/Riesel Base 5 Problem : Truthfulness of "largest known base 5 prime"

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numbermaniac
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Message 143718 - Posted: 27 Sep 2020 | 6:16:02 UTC

Pretty much every announcement about a new SR5 discovery will say that the number is "the largest known base 5 prime". Examples: 1, 2, 3.

However, didn't GCW discover 2805222*5^5610444+1, which, at 3.9 million digits, is a bigger base 5 prime than all of our SR5 discoveries? So why do we still say that every SR5 discovery is "the largest known base 5 prime"?
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Message 143720 - Posted: 27 Sep 2020 | 6:41:56 UTC

Is it even well-defined what a base 5 prime is?

In any case, even the example you mention has surely been beaten (Ryan Propper 6*5^6546983 + 1).

/JeppeSN

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Message 143743 - Posted: 28 Sep 2020 | 3:13:13 UTC - in response to Message 143720.

Is it even well-defined what a base 5 prime is?

Not sure, but I guess for our case I was considering only primes of the form k*5^n±1, since those are the base 5 primes that we search for.

In any case, even the example you mention has surely been beaten (Ryan Propper 6*5^6546983 + 1).


Yeah, that's true, I didn't think about that one from T5K. Doesn't this mean that none of our SR5 discoveries since that Propper prime were the largest known base 5 prime?
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Message 143750 - Posted: 28 Sep 2020 | 11:11:00 UTC - in response to Message 143743.

Is it even well-defined what a base 5 prime is?

Not sure, but I guess for our case I was considering only primes of the form k*5^n±1, since those are the base 5 primes that we search for.


Yes, but maybe we need a criterion like 5^n > k or something, otherwise there are so many primes that can be brought to that form.

In any case, even the example you mention has surely been beaten (Ryan Propper 6*5^6546983 + 1).


Yeah, that's true, I didn't think about that one from T5K. Doesn't this mean that none of our SR5 discoveries since that Propper prime were the largest known base 5 prime?


They need to remember to simply remove the statement about the SR5 being the largest known base 5 prime.

/JeppeSN

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Message 143757 - Posted: 28 Sep 2020 | 17:48:58 UTC - in response to Message 143750.
Last modified: 28 Sep 2020 | 17:50:02 UTC

Yes, but maybe we need a criterion like 5^n > k or something, otherwise there are so many primes that can be brought to that form.
If they can be written as 5^n I would say they are genuine. Isn't it convention to write a power with the base as small a value as possible? Smaller than 5 isn't possible, so all bases of form 5^n would be base 5 primes in my opinion.

I don't think there are that many cases anyway, only bases 25, 125, 625 are likely to encountered, I guess.

And arbitrary rules to keep the number of base 5 primes low are not a good thing to me.
____________
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Message 143758 - Posted: 28 Sep 2020 | 18:19:58 UTC - in response to Message 143757.

Yes, but maybe we need a criterion like 5^n > k or something, otherwise there are so many primes that can be brought to that form.
If they can be written as 5^n I would say they are genuine. Isn't it convention to write a power with the base as small a value as possible? Smaller than 5 isn't possible, so all bases of form 5^n would be base 5 primes in my opinion.

I don't think there are that many cases anyway, only bases 25, 125, 625 are likely to encountered, I guess.

And arbitrary rules to keep the number of base 5 primes low are not a good thing to me.


To explain what I mean, consider the forty-sixth Mersenne prime 2^42643801 - 1. This can be written on the form:

k*5^3 + 1

if you pick k correctly. So is 2^42643801 - 1 a base 5 prime?

/JeppeSN

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Message 143760 - Posted: 28 Sep 2020 | 18:41:44 UTC

Thanks, this will be fixed. I agree with JeppeSN that this is not a very precisely defined concept, but any reasonable definition (e.g. 5^n > k) would include both the GCW and the Ryan Propper prime. The only thing I don't like about that definition is that it also includes things like 993*10^1768283-1, which common sense would say is a base-10 prime and not a base-5 one.

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Message 143764 - Posted: 29 Sep 2020 | 5:57:52 UTC - in response to Message 143758.

Ah, I see. I always forget that possibility. So of course, you are right with the restriction. :)

Reminds of when I was pondering why 9 * 2^2n + 1 is a generalized Fermat prime...
____________
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Message 144027 - Posted: 9 Oct 2020 | 0:56:47 UTC - in response to Message 143760.

Thanks, this will be fixed. I agree with JeppeSN that this is not a very precisely defined concept, but any reasonable definition (e.g. 5^n > k) would include both the GCW and the Ryan Propper prime. The only thing I don't like about that definition is that it also includes things like 993*10^1768283-1, which common sense would say is a base-10 prime and not a base-5 one.

Perhaps the way to deal with that would be to extend the definition to say something like, "the base must be either 5 or an integer power of 5", which would allow things like SR5, and the GCW base 25 prime, but exclude base 10 primes like that one.
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Message 144029 - Posted: 9 Oct 2020 | 3:48:15 UTC - in response to Message 144027.

Perhaps the way to deal with that would be to extend the definition to say something like, "the base must be either 5 or an integer power of 5", which would allow things like SR5, and the GCW base 25 prime, but exclude base 10 primes like that one.

So a base-5 prime is defined to be a prime whose base is 5 (or a power of 5)? Sounds like a circular definition to me.

I really think it's futile to come up with a precise mathematical definition of "base-5 prime". For example, which of the following (composite) numbers would you consider to be "base-5"?

2^1 * 5^1000000 + 1
2^10 * 5^1000000 + 1
2^100 * 5^1000000 + 1
2^1000 * 5^1000000 + 1
2^10000 * 5^1000000 + 1
2^100000 * 5^1000000 + 1
2^1000000 * 5^1000000 + 1
2^1000000 * 5^100000 + 1
2^1000000 * 5^10000 + 1
2^1000000 * 5^1000 + 1
2^1000000 * 5^100 + 1
2^1000000 * 5^10 + 1
2^1000000 * 5^1 + 1

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Message 144035 - Posted: 9 Oct 2020 | 6:41:24 UTC - in response to Message 144029.

I really think it's futile to come up with a precise mathematical definition of "base-5 prime". For example, which of the following (composite) numbers would you consider to be "base-5"?

2^1 * 5^1000000 + 1
2^10 * 5^1000000 + 1
2^100 * 5^1000000 + 1
2^1000 * 5^1000000 + 1
2^10000 * 5^1000000 + 1
2^100000 * 5^1000000 + 1
2^1000000 * 5^1000000 + 1
2^1000000 * 5^100000 + 1
2^1000000 * 5^10000 + 1
2^1000000 * 5^1000 + 1
2^1000000 * 5^100 + 1
2^1000000 * 5^10 + 1
2^1000000 * 5^1 + 1


What is the largest known prime for which we are not really sure if it is base 5 or base 10 (a joke!)? /JeppeSN

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Message 144044 - Posted: 9 Oct 2020 | 11:58:23 UTC - in response to Message 144035.

I really think it's futile to come up with a precise mathematical definition of "base-5 prime". For example, which of the following (composite) numbers would you consider to be "base-5"?

2^1 * 5^1000000 + 1
2^10 * 5^1000000 + 1
2^100 * 5^1000000 + 1
2^1000 * 5^1000000 + 1
2^10000 * 5^1000000 + 1
2^100000 * 5^1000000 + 1
2^1000000 * 5^1000000 + 1
2^1000000 * 5^100000 + 1
2^1000000 * 5^10000 + 1
2^1000000 * 5^1000 + 1
2^1000000 * 5^100 + 1
2^1000000 * 5^10 + 1
2^1000000 * 5^1 + 1


What is the largest known prime for which we are not really sure if it is base 5 or base 10 (a joke!)? /JeppeSN


Let's see... Since there are an infinite number of primes, then the answer is infinity! (also a joke!)

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Message 144052 - Posted: 9 Oct 2020 | 15:46:24 UTC - in response to Message 144029.

I really think it's futile to come up with a precise mathematical definition of "base-5 prime". For example, which of the following (composite) numbers would you consider to be "base-5"?

2^1 * 5^1000000 + 1
2^10 * 5^1000000 + 1
2^100 * 5^1000000 + 1
2^1000 * 5^1000000 + 1
2^10000 * 5^1000000 + 1
2^100000 * 5^1000000 + 1
2^1000000 * 5^1000000 + 1
2^1000000 * 5^100000 + 1
2^1000000 * 5^10000 + 1
2^1000000 * 5^1000 + 1
2^1000000 * 5^100 + 1
2^1000000 * 5^10 + 1
2^1000000 * 5^1 + 1


Maybe say something like "base b prime" is a prime of the form p=k*b^n+/-1 with b>1 that maximizes b^n-k, gcd(k,b)=1? Not sure whether this makes sense
Then I think those numbers would be classified as
------------------------------ base 5
2^1 * 5^1000000 + 1
2^10 * 5^1000000 + 1
2^100 * 5^1000000 + 1
2^1000 * 5^1000000 + 1
2^10000 * 5^1000000 + 1
2^100000 * 5^1000000 + 1
------------------------------ base 10
2^1000000 * 5^1000000 + 1
------------------------------ base 2
2^1000000 * 5^100000 + 1
2^1000000 * 5^10000 + 1
2^1000000 * 5^1000 + 1
2^1000000 * 5^100 + 1
2^1000000 * 5^10 + 1
2^1000000 * 5^1 + 1

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Message 144055 - Posted: 9 Oct 2020 | 16:40:58 UTC - in response to Message 144052.

Maybe say something like "base b prime" is a prime of the form p=k*b^n+/-1 with b>1 that maximizes b^n-k, gcd(k,b)=1?

By that standard, every prime p would be a base-(p+1) prime, using k=n=1 and c=-1. More seriously, I think it would be perfectly reasonable to call e.g. 2^1000000 * 5^100000 + 1 = 5120^100000 + 1 a "base-5120" number rather than a base-2 number.

I guess my definition would be a bit more pragmatic: if you discover a prime using an LLR test in base b, or if you reasonably could have done so, then it's a base-b prime. (Or possibly a base-c prime where b is a power of c.)

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Message 144057 - Posted: 9 Oct 2020 | 16:57:17 UTC - in response to Message 144055.
Last modified: 9 Oct 2020 | 17:04:27 UTC

Oh, you're right. The way I classified the examples with my attempt on a definition was completely wrong haha, never mind

maybe then b = "largest factor with maximum multiplicity in p-1 (incl. composites)" yields my intended classification?

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Message 144059 - Posted: 9 Oct 2020 | 17:14:33 UTC

Note that gcd(k,b)=1 is a bad idea. Should be "b does not divide k" instead, as examples such as 5*10^511056 - 1 and 8*10^1715905 - 1 show. /JeppeSN

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Message 144060 - Posted: 9 Oct 2020 | 17:16:33 UTC - in response to Message 144057.

maybe then b = "largest factor with maximum multiplicity in p-1 (incl. composites)" yields my intended classification?

With the number 684*10^1127118 + 1, taking the base as b=2 gives a higher multiplicity than taking base b=10. /JeppeSN

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Message 144062 - Posted: 9 Oct 2020 | 17:32:06 UTC

Isn't the easiest definition the one already in place for Proth? As Jeppe wrote, b^n > k.

For example, which of the following (composite) numbers would you consider to be "base-5"?
All those where 5^n > 2^m. Since more talented people than me (not hard when it comes to math) in this forum apparently don't deem it that easy, where's the catch?


I you want a bit more rules you could say b has to be as large as possible. I.e. 5^21 wouldn't be base 5, but 125^7.
____________
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Message 144063 - Posted: 9 Oct 2020 | 17:55:47 UTC - in response to Message 144062.

where's the catch?

As hinted above, a number like 2^1000000 * 5^1000000 + 1 should probably be base 10 (so neither base 2 nor base 5). And Ravi explained why 2^1000000 * 5^100000 + 1 should likely be base 5120 (=5*2^10).

On the Top 5000, there are not many examples where there are two powers multiplied together in the same term. But I do see things like:
p = 2^14753 * 453931# * 453949^38004 + 1
Here, p-1 is easy to factor, but what is the "base"? 453949?

/JeppeSN

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Message 144073 - Posted: 10 Oct 2020 | 1:39:37 UTC - in response to Message 144029.

Perhaps the way to deal with that would be to extend the definition to say something like, "the base must be either 5 or an integer power of 5", which would allow things like SR5, and the GCW base 25 prime, but exclude base 10 primes like that one.

So a base-5 prime is defined to be a prime whose base is 5 (or a power of 5)? Sounds like a circular definition to me.

I really think it's futile to come up with a precise mathematical definition of "base-5 prime". For example, which of the following (composite) numbers would you consider to be "base-5"?

2^1 * 5^1000000 + 1
2^10 * 5^1000000 + 1
2^100 * 5^1000000 + 1
2^1000 * 5^1000000 + 1
2^10000 * 5^1000000 + 1
2^100000 * 5^1000000 + 1
2^1000000 * 5^1000000 + 1
2^1000000 * 5^100000 + 1
2^1000000 * 5^10000 + 1
2^1000000 * 5^1000 + 1
2^1000000 * 5^100 + 1
2^1000000 * 5^10 + 1
2^1000000 * 5^1 + 1


When I said "extend", that meant including the 5^n > k restriction as well, which would eliminate some of those possibilities. But yes, that does still leave some of the other problems that have been pointed out in this thread.
____________
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Message 144076 - Posted: 10 Oct 2020 | 5:45:12 UTC
Last modified: 10 Oct 2020 | 5:46:59 UTC

Ok, maybe like this:

b has to be as large as possible


I think that contains the b^n > k rule for ambiguous cases, while it allows numbers like 10001 * 5^2 to be a genuine base 5 number even though b^n < k. But it would force 10000 * 5^2 to be written as 25 * 100^2 which makes sense in my opinion.
____________
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Message 144077 - Posted: 10 Oct 2020 | 6:42:55 UTC - in response to Message 144076.

Ok, maybe like this:

b has to be as large as possible


I think that contains the b^n > k rule for ambiguous cases, while it allows numbers like 10001 * 5^2 to be a genuine base 5 number even though b^n < k. But it would force 10000 * 5^2 to be written as 25 * 100^2 which makes sense in my opinion.


The thread goes on forever, but the latter number is nicer as either 25 * 10^4 or 16 * 5^6. /JeppeSN

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Message boards : Sierpinski/Riesel Base 5 Problem : Truthfulness of "largest known base 5 prime"

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