# Finding the first digit of $2^{74207281}-1$ (new biggest prime record)

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'?

• By first do you refer to the left or to the right? – sinbadh Jan 20 '16 at 11:43
• @sinbadh Thank you, from the left! – mrprottolo Jan 20 '16 at 11:44
• 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"? – Arthur Jan 20 '16 at 11:52
• 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. – mrprottolo Jan 20 '16 at 11:56
• Also, in binary representation, that digit is $1$. :-) – 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.

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

First digits :

3003764180846061820529860983591660500568758630303014848439416933455477232190679942968936553007726883204482148823994267278352907009048364322180153481996522413722876843102133862845736663615066675321227728593598640577802568756477958658321420511711096358442629365726503872407101479826313204371431291121983921887612885039587719203550171864386658099542863444605366067617179336837496247567825783617310448839341553870852508685372972059312516068497815326704147449282948834494294439990037768310724968682506228660399788845410622342191545046452523868463034697248073341558528894973747787053275941448082695460497456828866626343377860615513544982943927889697172778141702478578408251738141699795297188313782581564608555984048010122779636641181623187402419844463395711475008938733504717522823092769609083682182574758579493336886487816470849356003894428166151012698929416209237005839204383031555766751286977273530159661985701199715089754997694301136325207049765960186628185272133382975016900338946922123296485757802701419640294542973795987529


Last digits :

66899171865137661519802116968797652141490953415212910857051284673598768690185271904942257300495816397525852539385053511561005650888650415738022663320276450183286594189674893176180397372990308168458435170835102089043676200149565779472442501535620998561290570036272472745402151726838956502770332764838294600205565782168765834170009350785028775004659016772336142517609382919278760066514029974961234186924815072110183841065704849092928560900884902955834170658601438843513844182462608969960624663300180408268566068553042312452267239351701254063934290920630706805289782939553784836331982240636162655281521278680346794370033755641577681719012062559469081273978710860347811180188837898128568440669359271612444713805577302483892184777905493456249144515504366735435257646973008855321674803866037094498725552912123074801792765597096176486305356033886997788467889060830923906229428002877708466815350114276229212218369040454779639313670134014480149404704116966334745646885160717774014762912462113646879425801445107393100212927181629335931494239018213879217671164956287190498687010073391086436351


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

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