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However, it turns out that if one of the sets, say A, is sufficiently “uniform” or “pseudorandom”, then one can always solve this Goldbach-type problem, regardless of what the other two sets are. This type of fact is often established by Fourier-analytic means (or by closely related techniques, such as spectral theory), but let me give a heuristic combinatorial argument to indicate why one would expect this type of phenomenon to occur.

quares Primes Lagrange’s four square theorem: For every positive integer N, there exists a pattern in of the form . Vinogradov’s theorem: For every sufficiently large integer N, there exists a pattern in of the form . Fermat’s two square theorem: For every prime number , there exists a pattern in of the form . Even Goldbach conjecture: For every even number , there exists a pattern in of the form . Fermat’s four square theorem: There does not exist any pattern in of the form with . Green-Tao theorem: For any , there exist infinitely many patterns in of the form with . Pell’s equation: There are infinitely many patterns in of the form . Sophie Germain conjecture: There are infinitely many patterns in of the form .

Checking the Odd Goldbach Conjecture Up to 1020