Denote by \(p_1 = 2 < p_2 = 3 < p_3 = 5 < \dots\) the ordered sequence of prime numbers.
Define the k-th primorial as
\[ P_k \;=\; \prod_{j=1}^{k} p_j . \]
For any even integer \(n \ge 6\) let
\[ G(n)\;=\;\bigl|\{(p,q)\in \mathbb P^{2}\mid p \le q,\; p+q = n\}\bigr| \]
be the Goldbach function, which counts the Goldbach partitions of \(n\).
An even integer \(N\) is called a flexible number if \(G(N) \ge G(m)\) for every even \(m < N\).
Sarko’s Conjecture states that all primorials are flexible numbers.
Sarko’s Conjecture. For every integer \(k \ge 2\) and for every even integer \(m \le P_k\),
\[ G(m)\;\le\;G(P_k). \]
No proof or counterexample is known.