Are the first 1,000 prime numbers enough to build every Goldbach number up to 9 digits long? I'm writing a basic computer program in which one of my requirements is to find the smallest pair of prime numbers that make up a Goldbach number (up to 9 digits long, non-inclusive). 
The user inputs any positive even number and I want the program to check if the summing parts are primes, but I want to maximize space so I want to only the smallest list of consecutive primes to check. 
pretty much I want to know what are the fewest number of primes I need to check?
The first 1000? the first 2000?
 A: Suppose the plan is to take a smaller prime from the given list, and to test the primality of the remaining difference by trial division.  Then we would need a list of primes that extends roughly to the square root of one billion ($10^9$).  That is, the list would need to go up to the prime 31607 (the largest prime less than the square root of one billion).
According to the Prime Pages at utm.edu, "There are 3,401 primes less than or equal to 31,607."  So if I understood your plan, you need a list of 3,400 primes (we will drop 2 as a prime because we will work only with odd primes).
Added: Of course typically for smallish trial division like we consider here, it's not horribly inefficient just to generate the superset of odd divisors consisting of $k \equiv 1,5 \mod 6$ on the fly.  So perhaps the question is what list of primes is need to guarantee the smaller prime of a "minimal Goldbach partition" is there (see @GerryMyerson).
There's a paper by Granville, te Riele, and van de Lune (1989) that conjectures the smaller prime needed to partition n is bounded above by a constant times $(\ln n)^2 \ln \ln n$.  Computational searching by Richstein (2000) showed that up to $n = 10^{14}$ the constant can be taken as 1.603, which would imply we only need a list of primes up to 2081, the 312th odd prime.
That's a far shorter list than one to be used for trial divisors as well.
A: 278 primes suffice; no primes greater than 1789 are needed for numbers below 1,847,133,842.
With 1000 primes you could go up to 121,005,022,304,007,024.
