Join PrimeGrid
Returning Participants
Community
Leader Boards
Results
Other
drummerslowrise

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

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"?
____________
2 PPSE & 3 SGS primes  


Is it even welldefined 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  


Is it even welldefined 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?
____________
2 PPSE & 3 SGS primes  


Is it even welldefined 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  

BurVolunteer tester
Send message
Joined: 25 Feb 20 Posts: 396 ID: 1241833 Credit: 154,178,093 RAC: 977,392

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.
____________
Primes: 1281979 & 12+8+1979 & 1+2+8+1+9+7+9 & 1^2+2^2+8^2+1^2+9^2+7^2+9^2 & 12*8+19*79 & 12^81979 & 1281979 + 4 (cousin prime)  


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 fortysixth 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  

Ravi FernandoProject administrator Volunteer tester Project scientist Send message
Joined: 21 Mar 19 Posts: 158 ID: 1108183 Credit: 9,356,373 RAC: 4,908

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*10^17682831, which common sense would say is a base10 prime and not a base5 one.  

BurVolunteer tester
Send message
Joined: 25 Feb 20 Posts: 396 ID: 1241833 Credit: 154,178,093 RAC: 977,392

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...
____________
Primes: 1281979 & 12+8+1979 & 1+2+8+1+9+7+9 & 1^2+2^2+8^2+1^2+9^2+7^2+9^2 & 12*8+19*79 & 12^81979 & 1281979 + 4 (cousin prime)  


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*10^17682831, which common sense would say is a base10 prime and not a base5 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.
____________
2 PPSE & 3 SGS primes  

Ravi FernandoProject administrator Volunteer tester Project scientist Send message
Joined: 21 Mar 19 Posts: 158 ID: 1108183 Credit: 9,356,373 RAC: 4,908

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 base5 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 "base5 prime". For example, which of the following (composite) numbers would you consider to be "base5"?
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  


I really think it's futile to come up with a precise mathematical definition of "base5 prime". For example, which of the following (composite) numbers would you consider to be "base5"?
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  

Scott BrownVolunteer moderator Project administrator Volunteer tester Project scientist
Send message
Joined: 17 Oct 05 Posts: 2255 ID: 1178 Credit: 10,043,808,751 RAC: 11,095,633

I really think it's futile to come up with a precise mathematical definition of "base5 prime". For example, which of the following (composite) numbers would you consider to be "base5"?
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!)
 


I really think it's futile to come up with a precise mathematical definition of "base5 prime". For example, which of the following (composite) numbers would you consider to be "base5"?
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^nk, 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
 

Ravi FernandoProject administrator Volunteer tester Project scientist Send message
Joined: 21 Mar 19 Posts: 158 ID: 1108183 Credit: 9,356,373 RAC: 4,908

Maybe say something like "base b prime" is a prime of the form p=k*b^n+/1 with b>1 that maximizes b^nk, 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 "base5120" number rather than a base2 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 baseb prime. (Or possibly a basec prime where b is a power of c.)  


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 p1 (incl. composites)" yields my intended classification?  


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  


maybe then b = "largest factor with maximum multiplicity in p1 (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  

BurVolunteer tester
Send message
Joined: 25 Feb 20 Posts: 396 ID: 1241833 Credit: 154,178,093 RAC: 977,392

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 "base5"? 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.
____________
Primes: 1281979 & 12+8+1979 & 1+2+8+1+9+7+9 & 1^2+2^2+8^2+1^2+9^2+7^2+9^2 & 12*8+19*79 & 12^81979 & 1281979 + 4 (cousin prime)  


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, p1 is easy to factor, but what is the "base"? 453949?
/JeppeSN  


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 base5 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 "base5 prime". For example, which of the following (composite) numbers would you consider to be "base5"?
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.
____________
2 PPSE & 3 SGS primes  

BurVolunteer tester
Send message
Joined: 25 Feb 20 Posts: 396 ID: 1241833 Credit: 154,178,093 RAC: 977,392

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.
____________
Primes: 1281979 & 12+8+1979 & 1+2+8+1+9+7+9 & 1^2+2^2+8^2+1^2+9^2+7^2+9^2 & 12*8+19*79 & 12^81979 & 1281979 + 4 (cousin prime)  


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  

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