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I was just reading Is every positive nonprime number at equal distance between two prime numbers? (current hot topic) and was reflecting on the fact that computing security (cryptography) is based around the use of large prime numbers (see: Why are very large prime numbers important in cryptography?).

The answer for the above-mentioned question suggests that the Goldbach Conjecture says that every non-prime positive number is positioned equidistant between two prime numbers.

For the purposes of this question, I'll assume that statement is true (I have no prior knowledge of Goldbach or his/her conjecture).

If the Goldbach Conjecture is true, does it make it easier to find large prime numbers?

For example, I could take any very large number at random and then look at every number below it, find a prime and then work out of the opposite number is also a prime (or something along those lines). In my mind, it's almost as though the assumption would give me a starting point to find an even larger prime...

I expect I'm not the first person to ask this (and if I am, I've probably missed something somewhere..), but I can't find a similar question here :)

Thanks in advance

Community
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m-smith
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    How is that going to be efficient? Given number $n$, you'd have to walk through the primes less than $n$, which, when $n$ is, say, $100$ digits, is going to be too many primes to check. And you are effectively skipping candidates, because Goldbach doesn't say that $n-k$ is prime if and only if $n+k$ is prime. – Thomas Andrews Nov 16 '11 at 14:54

2 Answers2

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Finding primes of the size wanted for cryptography isn't hard. The prime number theorem says that a "random" number $n$ has one chance in $\ln n$ of being prime. For a $1000$ bit number, this is about $1$ in $700$. If you only try numbers congruent to $1$ or $5 \pmod {6}$, you get another factor $3$, so you only have to try a few hundred before you find one. How to check is described here. The celebrated prime numbers that are found are much larger.

Ross Millikan
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Technically it could, in some exceptionally unlikely scenario were you have a large even number $2n$, and you find that all $2n-p_i$ are all composite with small prime-factors except for one small $i$.

But seriously the only information you gain is that atleast one of $2n-p_i$ is prime, but heuristically this is to be expected 99.999% anyways so you gain really nothing except if above magic conspiracy took place.

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