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How many digits these primes have

The known values of n that make 3*2^n-1 prime are:

1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676, 14898, 18123, 18819, 25690, 26459, 41628, 51387, 71783, 80330, 85687, 88171, 97063, 123630, 155930, 164987, 234760, 414840, 584995, 702038, 727699, 992700, 1201046, 1232255, 2312734, 3136255, 4235414

Viewing the decimal expansion that most humans use, the sequence of primes starts as:

3*2^1-1= 5
3*2^2-1= 11
3*2^3-1= 23
3*2^4-1= 47
3*2^6-1= 191
3*2^7-1= 383
3*2^11-1= 6143
3*2^18-1= 786431
3*2^34-1= 51539607551
3*2^38-1= 824633720831
3*2^43-1= 26388279066623
3*2^55-1= 108086391056891903
3*2^64-1= 55340232221128654847
3*2^76-1= 226673591177742970257407
3*2^94-1= 59421121885698253195157962751
3*2^103-1= 30423614405477505635920876929023
3*2^143-1= 33451117797795934712303577408972542258970623

The numbers grow and are too long to display here; the largest known 321 prime has 1,274,988 decimal digits.

In binary the first few 321 primes look like this:

3*2^1-1= 101
3*2^2-1= 1011
3*2^3-1= 10111
3*2^4-1= 101111
3*2^6-1= 10111111
3*2^7-1= 101111111
3*2^11-1= 1011111111111
3*2^18-1= 10111111111111111111
3*2^34-1= 101111111111111111111111111111111111
3*2^38-1= 1011111111111111111111111111111111111111
3*2^43-1= 101111111111111111111111111111111111111111111

The largest known 321 prime when written as a binary number consists of a one and a zero followed by 4,235,414 ones.


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