
#19
Jul708, 10:13 PM

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#20
Jul808, 12:41 PM

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Why is the mathematical world so focused on prime numbers, why not the general case, to find all numbers with n prime factors?




#21
Jul808, 12:52 PM

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2. Many encryption algorithms rely on the hardness of factoring large semiprimes; factoring primes is, of course, trivial. 



#22
Jul808, 12:57 PM

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#23
Jul908, 12:57 AM

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Kurret, A few more thoughts, adding to what Greathouse said. One reason there is a "focus" on "prime numbers" these days is the relation with the Riemann hypothesis. For example, if we can show that the error term in the prime number theorem grows at the order of the square root of n, then we will know that the Riemann Hypothesis is true. (ref: Barry Mazur's "error term" article in this issue of the BAMS; the equivalence was shown about 100 years ago). So, this fact reduces your question to "why is there so much interest in the Riemann hypothesis"? Well, first of all, the RH is a very old problem. Second, if it is true, many intereesting and beautiful facts about nature will follow. (My viewpoint is that interesting mathematics describes interesting things about nature.) Third, everything that is known about the RH gives the impression that we a re very close to solving it. Fourth, it's been that way for about 100 years now, so why haven't we solved it? (Riemann even wrote down an exact formula for the error term in the prime number theorem. It would seem that all we have to do is analyze that formula.) Fifth, numerical studies of the behavour of the zeroes of the zeta function on the critical line indicate that current techniques are inadequate to solve it. The behaviour of those zeroes is just too strange and too sbutle to get a handle on. Therefore, a solution would likely result in a significant advance in our understanding of mathmatics in general. Sixth, there is a million dollar prize for the person who resolves it. Seventh, the name of the person who solves it is very likely to go down in history for a very long time. On the other hand, as Greathouse pointed out, the question you ask is really of very high interest to today's mathematicians. One reason for that is IF we could find an effectgive algorithm to answer your question, THEN we would be a lot closer to finding an effective algorithm to factoring the product of two large primes in a short period of time. This would break RSA cryptography. It is highly likely that the proof would carry over to break discrete logarithm cryptography. It is quite possible that the proof would allow us to break elliptic curve cryptography. The underlying mathematical metaprinciple that you are touching on is that it is a lot easier to ask hard questions than it isw to ask "good" quesitons. A "good" question is a question that we have a hope of solving; that we feel that if we worked on it, we should be able to come up with something. A small group of very good mathematicians are currently investigating the properties of numbers that are the product of a large number of small primes. The fact that that investigation is difficult (but doable) underlines just how hard the question that you ask probably is. Another good example of a "hard" question is the question of whether or not there are an infinite number of Mersenne primes. For that one, there even exists a heuristic argument that there are an infinite number and it yields an asympototic estimate for how many there are! And yet, as far as I know, nobody has any idea about how to get started on actually proving that there are an infinite number of them. Richard Guy also said this in his book "Open Problems in Number Theory" or something like that. Oh, oh, oh, except the female star of the science fiction movie called "Proof." The "science fiction" in the movie is that she resolved the question about the finiteness of the Mersenne primes. DJ P.S. I welcome corrections, or even "that doesn't sound right" comments. That's how I learn. 



#24
Jul1108, 10:41 AM

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I have good news the discoverer of this formula is going to publish a book that contains many results like defining the set of twin prime numbers a formula about producing nth prime and many other top results .




#25
Jul1108, 02:17 PM

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What, then, are these "other top results"? 



#27
Jul1108, 11:54 PM

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why you don not respect others attempt
have you seen this book why you are so judgemental 



#28
Jul1208, 01:39 PM

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#29
Jul1208, 03:16 PM

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There is certainly no point in making qualifying statements about a claim that has not been presented in the necessary detail.




#30
Jul1208, 07:49 PM

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#31
Jul1308, 01:47 AM

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I assume you guys are being sarcastic right?
There is no formula (function) which, given an arbitrary input will guarantee a prime output, let alone, give you the primes in order. You can work recursively, (ie, the sieve of Eratosthanes) but its not really a formula, it just knocks out the composites one prime factor at a time. Nor has it been proven (conclusively) that there are an infinitude of twin primes. 



#32
Jul1308, 09:31 AM

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Ruiz, S. M. "The General Term of the Prime Number Sequence and the Smarandache Prime Function." Smarandache Notions J. 11, 5961, 2000. [tex]p_n=1+\sum_{k=1}^{2(\lfloor n\ln n\rfloor+1)}\left[1\left\lfloor\sum_{j=2}^k1+\lfloor s(j)\rfloor}/n\right\rfloor\right][/tex] where [tex]s(j)=\frac{\sum_{s=1}^j\left(\lfloor j/s\rfloor\lfloor(j1)/s\rfloor\right)2}{j}[/tex] There are many more examples, some simpler and some more complex. Some give all primes in order, some give all distinct primes, some give the count of primes through n, etc. 



#33
Jul1408, 04:58 PM

P: 5

f(x) = x ; if x is prime does too
let me rephrase, there is no function that is not based on recursion or more fundamentally on a sieve process. Making a function that step by step knocks out the composites is not exactly what I meant. "My bad" on the linguistics there, sorry 



#34
Jul1408, 06:42 PM

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#35
Jul1408, 07:55 PM

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I'm not trying to be stupid, i really think this is a semantic / linguistic confusion, not an argument.
Looks to me like wilsons theorem is a test, not a function. But, anything with factorials, list all previous values (factors). So the Smarandache Prime Function essential shows that if you list out all of the numbers multiplied together, then there are sufficient factors to generate every number up to the current number, so if its prime, then the test holds. straight from wolframs site: /quote For example, the number 8 does not divide 1!, 2!, 3!, but does divide 4!=4·3·2·1=8·3, so mu(8)=4 /endquote So mu(5) will be 5 because 5 is prime, because none of the previous factors give multiples of five, this is essentially saying f(x) = x is prime, if x is prime All I mean is, there is no function F(M)=P for some input M that guarentees P is prime, and M is any single valued input, be it real, integer, complex or quaternian. If there where, it would be bloody famous and there'd be no need for prime testing algorithms. Especially if it was an invertable function! But think of it, if you had such a function and you wanted to test if a REALLY BIG NUMBER was prime. You start plugging in inputs whose outputs are close to your REALLY BIG NUMBER until you get the result. It wouldn't even have to be a continuous function, just ordered. So I think i am using a much more limited definition of function than you are. The sieve of eratothanes can be written in function form, but its still the sieve Any formula that uses Sigma summation (sorry, havn't used tex in years, forgot syntax) is an algorithm, which I agree, there are lots of. And they run through countless computations to generate only a few outputs, thus making them inneffective for very large primes I have a nice algorithm that uses the symmetry of p# and it makes the computer find primes very quickly, but it is not a function. (if your interested i'll show you, i'd like to know if its well known or perhaps something that hasn't been investigated too much anyway) Hope this clarifies 



#36
Jul1408, 09:15 PM

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Given the usual definition of 'function', it's obvious that there exist all sorts of functions defying your limitations. For example, there is the nth prime function. Explicitly, it's a function of the positive integers satisfying p(1) = 2 and p(n+1) = smallest prime exceeding p(n). And there are subclasses of functions which we can prove other theorems about. For example, every (nonconstant) polynomial function with integer coefficients must take on at least one nonprime value. Similarly, if f is such a polynomial, and it satisfies f(p)=0 for every prime p, then f is the zero polynomial. On the other hand, we have Matiyasevich's theorem which implies that there exists a polynomial g in many variables with integer coefficients for which [tex]g(p, x_1, x_2, \cdots, x_k) = 0[/tex] has an integer solution for the x's if and only if p is prime. If you want to say interesting things about some subclass of functions, you are going to have to find some way to express precisely what sorts of functions you want to talk about, otherwise your posts won't really have much content. 


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