I heard that the record for finding the largest prime number was broken a few days ago with the following Mersenne prime $$2^{74207281}-1$$ also called $M_{74207281}$. Now my question is: it is possible to compute the first digit (from the left) of this number 'by hand'?

  • $\begingroup$ By first do you refer to the left or to the right? $\endgroup$ – sinbadh Jan 20 '16 at 11:43
  • $\begingroup$ @sinbadh Thank you, from the left! $\endgroup$ – mrprottolo Jan 20 '16 at 11:44
  • $\begingroup$ When saying 'by hand', would you allow using normal (scientific) calculators, the kind that does logarithms and goes up to $10^{99}$ (as opposed to a full computer and dedicated software capable of representing all the 22 million digits)? Or do you really mean "with pen and paper"? $\endgroup$ – Arthur Jan 20 '16 at 11:52
  • $\begingroup$ Well, the ideal thing would be only using pen and paper, but I realize that this would be too long, so we are allowed to use a standard scientific calculator. $\endgroup$ – mrprottolo Jan 20 '16 at 11:56
  • 1
    $\begingroup$ Also, in binary representation, that digit is $1$. :-) $\endgroup$ – Asaf Karagila Jan 22 '16 at 21:45
? 10^(frac(p*log(2)/log(10)))
%81 = 3.003764180846061820528342824

So, the first digits are $3003764180846061820...$

To calculate the first digit by hand, you have to calculate $74207281\times log_{10}(2)$ upto $4$ digits after the comma and to take the fractional part.

Then you have to calculate $10$ to the power of that number and truncate. Could be done by hand, but would be difficult.

  • $\begingroup$ @OP if you want more, here are a few additional ones: 30037641808460618205298609835916771802 $\endgroup$ – mrf Jan 20 '16 at 11:55
  • $\begingroup$ I know, I noticed it when I used a higher precision. $\endgroup$ – Peter Jan 20 '16 at 11:56
  • $\begingroup$ @mrf: Yours seem to be off too, with two independent sources I get $30037641808460618205298609835916605005\dots$ $\endgroup$ – gammatester Jan 20 '16 at 12:00
  • $\begingroup$ $$22\ 338\ 618$$ digits. I am curious how long the double-check will take. $\endgroup$ – Peter Jan 20 '16 at 12:02
  • 1
    $\begingroup$ PARI can calculate that number completely! The digit sum is $100\ 537\ 543$. $\endgroup$ – Peter Jan 20 '16 at 12:22

First digits :


Last digits :


protected by Asaf Karagila Jan 22 '16 at 21:43

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.