Does Zhang's work imply that the next greatest prime number is necessarily less than $2^{57,885,161}-1+ 70000000$?

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Actually the greatest known prime is $2^{57,885,161}-1$. Zhang's work proved that the interval between pairs of prime numbers is limited to $70,000,000$. Does it imply that the next greatest prime number is necessarily less than $2^{57,885,161}-1 + 70000000$?

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No, it implies that there are always prime numbers that are separated by 70,000,000 after any given prime that you specify. There may not be another pair of primes with a separation of 70,000,000 until after the number $10^{10^{100}}$, who knows, but there certainly exists an infinite number of pairs of primes after $10^{10^{100}}$ with a difference of less than 70,000,000.