A Question About Prime Numbers and Goldbach's Conjecture

  • #1
I know that one of Goldbach's conjectures is that every even number is the sum of 2 primes.

So, I was wondering if there was a definite, largest prime number ever possible. I know that as a number gets larger, there are more numbers that can be tried to divide it (At least I think so), and I was wondering if there was a point where numbers got so large that there is a single highest prime number, and there can be no higher number that can't be divided by itself or 1.

So, we can probably do something with this. Doing so, we take an obscenely large number, much larger than the largest prime, say 20 times larger, for good measure, and then, we would not be able to make that even number by adding 2 primes to it, it would be impossible, if there is a largest prime.

If this is true, then I think that the version of Goldbach's conjecture stated will be disproven.

Please forgive me if anyone asked this before, I have not taken any classes on number theory yet, I only just finished Algebra 1 in high school, so I o not know much about it. But it just sparked in my mind, and I thought I might give a shot at seeing what people think about it. And, forgive me if I got anything wrong, though I don't think I did, since most of what I said were mostly questions.
 
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Answers and Replies

  • #2
mathman
Science Advisor
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One of the basic theorems of number theory (dating back to Euclid) is that there are an infinite number of primes.

Basic idea of proof (by contradiction) - assume finite list of primes, multiply them all together and add 1. Result is not divisible by any of the numbers on the list, so must be prime or divisible by a prime not on the list.
 
  • #3
Thank you for that explanation :D
 
  • #4
Below is a link to a page with 5 different proofs of the infinitude of the primes...

•Euclid's Proof (c. 300 BC)
•Furstenberg's Topological Proof (1955)
•Goldbach's Proof (1730)
•Kummer's Restatement of Euclid's Proof
•Filip Saidak's Proof (2005)
http://primes.utm.edu/notes/proofs/infinite/

Note that one of the proofs is by Goldbach and involves Fermat Numbers although offhand I don't see how that would directly relate to the Goldbach Conjecture other than to make clear that which is already clear by Euclid's proof, namely that one is dealing with an infinite number of cases.
 
  • #5
unfortunately, there are infinitely many primes.
however we have a good idea of how many there are up to a certain value.
number of primes up to n = n/ln(n)
to put it another way, perhaps easier for you to understand...
imagine we continuously multiply by the number 3,
1, 3, 9, 27, 81 and so on.
then imagine we put numbers in a pile, and increase the pile size by 1 each time we pass that number.
Code:
1 2 3 5 7 9  12 15 18 21 24 27 31
    4 6 8 10 13 16 19 22 25 28 32
          11 14 17 20 23 26 29 33
                            30 34
and so on.
the number of primes in each pile will be on average, 1. (though the multiplication should be closer to 2.718)
 
  • #7
144
0
Hi, try to read the following paper related to the strong Goldbach's Conjecture:

http://arxiv.org/abs/1208.2244

Hi Marco39,

I have had a first look at your paper "Some considerations in favor of the truth of
Goldbach’s Conjecture".

The title is somewhat vague and modest, and the first sections rather elementary. Then jumping to your one and only theorem, and skipping everything but the first and the last line of its statement, I read what seems to be a claim of Goldbachs Conjecture.

Before I go ahead and look for a possible error in your argument, can you please state in more explicit terms, that you think you have a proof of Goldbach. Thanks.
 

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