Find x(number theory)

[xshadysaysx]

Lets have a nice(and easy) question on my first post

 

Find all [math]x[/math] such that [math]x^2+2[/math] is a prime number.

[EvoN1020v]

You can find the primes using The Sieve of Eratosthenes.

 

I already tried this experiment (1 to 100), and it did work. You should get: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

 

Do you want the general formula for x for infinite amounts of prime numbers?

[the tree]

Wooh, finally someone new who actually bothers to use the forum properly. For interest value, Wikipedia has an article on the Sieve of Eratosthenes with a cute animation.

[woelen]

This is not a simple question at all. Of course, by means of sieve methods, you can enumerate all primes of the form x*x+2, and hence find x. Some possible values for x are: 0, 1, 3, 9, ... However, there is not a closed arithmetic or analytic expression for x, e.g. as function of an index (natural number).

There also is not a closed arithmetic or analytic expression for the prime numbers themselves, i.e. for the question: find all x, such that x is prime.

[the tree]

I suppose, given a sequence of primes it'd just be a case of [math]\sqrt{a_{n}-2}[/math], but alas, no such sequence is defined.

[the tree]

Of course you could do it (at least up to a finite value of [math]x[/math]), but that's not really the point.

s/he will want something that'll stand up in a mathematics exam.

[EvoN1020v]

Let's face it: There's no magic equation that'll tell you all the primes. It remains as one of the most famous unsolved mathematics of our time.

[matt grime]

I think I must disagree with that statement. We have a very simple way to define the n'th prime in a simple function (or to encode them in an equation) it's just that there is no nice way to evalute it in any reasonable time for any reasonably large n. There is a difference between a function, or equation, and actually evaluating that equation/function at a point. It is a subtle difference but one that is important.

 

Let's think on an example:

 

Define a_n by a_1=2, and a_n is the smallest integer greater than a_1,..,a_{n-1} not divisible by any previous term.

 

Define b_n by b_n=sin(n).

 

Find a_{10^30} and b_{10^30}.

 

The question here is what do we mean by "find"? a_(10^30) is the 10^30'th prime number, but I can't write that out as a decimal expansion. But then I can't write out b_(10^30) either as a decimal expansion (it is an irrational number in all likelihood) but you will probably happily accept sin(10^30) as "the answer", so why is one symbol better than the other? Sure I can find a huge number of the digits of sin(10^30), and very quickly, but I can never write them all out, I might even be able to give some kind of formula for them, whereas, although it takes a huge amount of time to compute the a_n for large n I could in theory, if the universe didn't end work out all the decimal digits for any one of them.

 

There are plenty of functions whose positive values are exactly the set of prime, or whose roots occur at exactly the set of primes, for instance.

 

And if anyone wants to say 'yeah, well, if there is a magic equation then what's the 3 trillionth prime number?' then you've missed the point of this post. I can no more work out large primes than I can large terms of te Fibonnacci sequence; it is just computationally infeasible, the difference is that one becomes infeasibly more quickly than the other; just because there is a formula for the n'th fibonacci number doesn't really mean that we have a magic equation that pops out any term your care to name, and this is all about the mathematical interpretation of inherently non-mathematical labels, like 'magic', so this is a statement of opinion not an assertion of fact.

 

I would have been happy with you saying there was not 'a magic algorithm' to spew out the primes in milliseconds.

 

Here is a function all of whose zeroes are exactly the set of primes, by the way:

 

[math] f(x):= \prod_{p}(1-\frac{x}{p^2}) [/math]

 

Of course if we could find a nicer expression for that (that didn't involve p explicitly) and a quick way to find roots we'd be laughing.