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If a prime gap $g(p)$ is the first occurring prime gap of it's size, does this imply that it is also the largest gap below $p$? In other words, is the set of first occurring prime gaps contained completely within the set of maximum prime gaps below $p$?

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No. $(149,139)$ is the first pair of consecutive primes whose difference is $10$, but $$127-113=14>10. $$

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