Exploring Prime Numbers: Digit Counts

[abertram28]

let me refine the light of the context of my statement. obviously there are trivial solutions. its meaningless as a generator to use the algorithm "2" to generate primes.

the context was the generation of primes. if an algorithm tests for primality, reguardless of its method, it isn't generating primes, but generating numbers which may be prime. then the test is a second algorithm. to include a primality test in an algorithm for generating primes is pretty meaningless. any algorithm that tests for primality is automatically a "prime generating algorithm" according to you. well, it might be, but its a useless one, as it is neither fast nor easy.

so, just because one has a method for testing primes, in my humble opinion, does not give a prime number generator. the number generating portion of the algorithm is all that's concerned when the words NUMBER GENERATING ALGORITHM are used. NOT a number testing algorithm. the generating portion of all of these algorithms is NOT producing strictly primes, in any form. if the generation portion has the capability of producing a composite number, the generator isn't a prime number generator, but a prime canidate generator. if you want to say that a 'generating' algorithm is one that includes a 'testing' algorithm, go right ahead. generally, i said there are no algorithms that always give primes. in context, i meant useful ones that don't involve actual testing of a large portion of the generated numbers, that is, numbers generated by the generator portion of the algorithm, or the generator algorithm. i should have said that there are no generation algorithms for primes. because there arent. only testing ones. if one combines both, it still relies on testing primality, which isn't an easy task. generation algorithms would not eliminate any generated number as being composite. can you name any algorithm that doesn't directly test a number for primality? algorithms that involve testing, to me, are composite algorithms, using two very separate algorithms, the generators and the testers. the two need not be connected. any generator could be used with any testing algorithm, and so, i see them as seperate. that's what i meant by calculate prime, not just 'find' a prime. brute force solutions are for babies.

 

[master_coda]

so, just because one has a method for testing primes, in my humble opinion, does not give a prime number generator.

Your opinion of what it means for something to be a prime number generator is irrelevant. Any procedure that enumerates the prime numbers is a prime number generator; it doesn't matter how the procedure operates, or if it's inefficient. This is standard terminology. Definitions don't get changed every time someone complains that they should be different.


You seem to be interested in having a prime number generator faster than one that just interates over the integers and tests them for primality. So ask about that, instead of trying to change the meaning of "prime number generator" to exclude algorithms that aren't fast enough for you.

 

[abertram28]

i wasnt trying to exclude ones that arent fast enough. i really don't care about the issue personally, but judging by the few of you who are posting, you must have some personal stake in this. why the attitude? did i step on someones toes in my first post? you could just kindly correct my terminology if you wanted.

any procedure that enumerates the prime numbers, while simple in defintion, is contextually meaninless. sure, there are plenty of useless algorithms for returning "2" no matter what you do. but they are contextually and physically meaninless, whether they fit the given definition. the defintion of generate has nowhere in it the "exhaustive exclusion" methods that testing algorithms have. so by definition, a generator should be the origin of the number, and the origin of the these numbers is NOT anywhere in the testing part of the algorithm. can you not admit that there are only one type of algorithm and that it is an algorithm for testing primality, not really for more specifically generating primes? the origin of the numbers, or the algorithm for generating numbers to be tested by the testing portion of the algorithm is more fitting of the defintion of "generator", so please link to your math dictionary so i can be more correct in terminology next time.

mathwonk was the one who claimed these methods to be fast and useful for generating large primes, which they certainly are not. while i have a passing interest in the topic, ill kindly use a lookup table if i need to use large primes for anything, rather than tediously generate them by exclusion algorithms.

what happened to the nice people here? did someone jade you all and make you necessarily evil? correcting someones terms is so commonplace, it shouldn't be done with such tone. i have no evil intent, so please don't make assumptions about my "seeming interest". i have none after these exchanges.

let me say, honestly, sorry if i hurt anyones feelings. maybe feelings arent the best thing to bring into a discussion like this. I am not trying to redefine algorithm, or any other term. i thought maybe expressing my opinion would have been a good way to get it exactly clear. we can drop this part of the thread and let it continue normally... jeesh.

and, btw, there is really no such thing as a prime number generator through enumeration. there isn't even an enumeration algorithm for generating the integer set, so don't nitpick definitions too much. no enumerative process can create an infitine set, by definition, the process must take finite steps, and its quite obvious this generation of a complete set with infinite unique members cannot take place in finite steps.

 

[CrankFan]

i wasnt trying to exclude ones that arent fast enough. i really don't care about the issue personally, but judging by the few of you who are posting, you must have some personal stake in this. why the attitude? did i step on someones toes in my first post? you could just kindly correct my terminology if you wanted.

What's irritating people is that you obviously don't know what you're talking about and even after it's been pointed out that you're wrong, you carry on as if you've been right all along.

Here, let me use an analogy, maybe this will help you see what's going on.

Just because you might mean something different than everybody else when you use the word "flat" doesn't mean that it's an acceptable thing for you to say that "the Earth is flat" on a scientific forum, and then for you to act surprised when everybody jumps into tell you that it's wrong.

Saying that "the Earth is flat" is just as ridiculous a thing to say as "there are no algorithms which generate prime numbers."

 

[master_coda]

and, btw, there is really no such thing as a prime number generator through enumeration. there isn't even an enumeration algorithm for generating the integer set, so don't nitpick definitions too much. no enumerative process can create an infitine set, by definition, the process must take finite steps, and its quite obvious this generation of a complete set with infinite unique members cannot take place in finite steps.

It doesn't matter if we can't produce a listing of the entire set in a finite amount of time; we don't have anywhere to store an infinite amount of data anyway, so we don't want to produce an infinite set. All we need is some algorithm which is guaranteed to produce any specific member of a set in a finite amount of time.


As for my tone...if it bothers you that I called your opinion irrelevant, that's unfortunate, but it's true. I'm happy that you can admit that your complaint was only based on your personal opinion, but the impression I got from your post was that you thought saying that this was your opinion somehow lent more weight to what you were saying. If it bothers you that people would attack you personal opinion, then don't post your opinion on a discussion board.

 

[mathwonk]

I suggest that this thread has become pointless.

 

[master_coda]

I suggest that this thread has become pointless.

Agreed. It's turned into bickering over terminology.

 

[slackr007]

I had trouble reading all three pages of this thread, but are you guys saying that prime numbers come in arbitrarily long, finite arithmetic progressions where it isn't always possible to predict the next number?

 

[Leopold Infeld]

I had trouble reading all three pages of this thread, but are you guys saying that prime numbers come in arbitrarily long, finite arithmetic progressions where it isn't always possible to predict the next number?

Can the 'guy' who said "the prime numbers come in arbitrarily long, finite arithmetic progressions" pls tell me where u got that from? It would also be nice if you could tell me who proved that. Because I've never heard that prime numbers 'do' have any pattern whatsoever.

 

[matt grime]

Tao and Green proved recently that there are arbitrary arithmetic progressions of primes. Which, for slackr007 is exactly one of the things you'd expect to find in "random" behaviour. Just because we know there is such an A.P. of some length with some common difference doesn't tell us where to find it. Indeed, I believe that the bounds on where to look are so astronomically big that they are of absolutely no use in computation whatsoever.