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
Best Answer
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.