Climbing High for Primes

Bertrand's Postulate states that for any integer $n>3$ there exists at least one prime number between $n$ and $2n$ .

If we let $n$ be $2^{74207281}-1$ , then we know there must be a prime between $2^{74207281}-1$ and $2\times\left(2^{74207281}-1\right)=2^{74207282}-2$ .

Since $2^{74207282}-2$ is less than $2^{77232917}-1$ , we know that there is at least one prime between $2^{74207281}-1$ and $2^{77232917}-1$ , and so the answer is no , $2^{74207281}-1$ is not the largest prime less than $2^{77232917}-1$ . In fact, since the larger number is roughly $2^{3,025,636}$ times larger than the smaller number, we can repeat Bertrand's Postulate for each additional power of 2, so that there are at least $3,025,636$ primes in between them!

The reason we have not found these primes is because it is easier to look for Mersenne primes, which similarly to the primes in the question are 1 less than a power of 2. Mersenne numbers have a relatively simple test to determine whether they are prime, and so now large primes are found by a collaborative project called the Great Internet Mersenne Prime Search .