Riemann Hypothesis implies Odd Number is Sum of at most 5 Primes

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Let the truth of the Riemann Hypothesis be assumed.

Let $n$ be an odd integer.

Then $n$ is the sum of at most $5$ primes.


Historical Note

When this result was cited by David Wells in the $1997$ edition of Curious and Interesting Numbers, 2nd ed., this was the limit of knowledge.

However, Terence Tao has since come up with a proof of this result which does not require the Riemann Hypothesis.