In his proof of the ternary Goldbach conjecture, H.A. Helfgott says that it has been verified that every odd number less than $N_0 = 10^{30}$ is the sum of at most 3 primes. How would one verify this up to $N_0$ in a reasonable amount of time. This is because even if one could verify in $O(1)$ time whether or not a number is the sum of 3 primes, to verify it until $10^{30}$ would still take a very long time.
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See http://arxiv.org/pdf/1305.3062.pdf for a detailed description of how the verification was obtained. It's pretty heavy though, as can be expected because naive verification would take far too long.
