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Question Number 56660 by tanmay.chaudhury50@gmail.com last updated on 20/Mar/19

Commented by kaivan.ahmadi last updated on 20/Mar/19

can you proof it sir?

$${can}\:{you}\:{proof}\:{it}\:{sir}? \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 21/Mar/19

i think to prove 2^(prime number) −1=prime number we need f(n) which is prime 2^(f(n)) −1=prime number to my knowldge no such general formula f(n) derived till now which describe prime number.

$${i}\:{think}\:{to}\:{prove} \\ $$$$\mathrm{2}^{{prime}\:{number}} \:−\mathrm{1}={prime}\:{number} \\ $$$${we}\:{need}\:{f}\left({n}\right)\:{which}\:{is}\:{prime} \\ $$$$\mathrm{2}^{{f}\left({n}\right)} −\mathrm{1}={prime}\:{number} \\ $$$${to}\:{my}\:{knowldge}\:{no}\:{such}\:{general}\:{formula}\:{f}\left({n}\right) \\ $$$${derived}\:{till}\:{now}\:{which}\:{describe}\:{prime}\:{number}. \\ $$$$ \\ $$

Commented by mr W last updated on 21/Mar/19

next one: 23 is prime 2^(23) −1=8388207=47×178481 ⇒no prime next one: 29 is prime 2^(29) −1=536870911=233×1103×2089⇒no prime therefore 2^(prime) −1 =_(but not always) ^(maybe) prime

$${next}\:{one}: \\ $$$$\mathrm{23}\:{is}\:{prime} \\ $$$$\mathrm{2}^{\mathrm{23}} −\mathrm{1}=\mathrm{8388207}=\mathrm{47}×\mathrm{178481}\:\Rightarrow{no}\:{prime} \\ $$$${next}\:{one}: \\ $$$$\mathrm{29}\:{is}\:{prime} \\ $$$$\mathrm{2}^{\mathrm{29}} −\mathrm{1}=\mathrm{536870911}=\mathrm{233}×\mathrm{1103}×\mathrm{2089}\Rightarrow{no}\:{prime} \\ $$$${therefore} \\ $$$$\mathrm{2}^{{prime}} −\mathrm{1}\:\:\underset{{but}\:{not}\:{always}} {\overset{{maybe}} {=}}\:\:\:\:{prime} \\ $$

Commented by mr W last updated on 21/Mar/19

if 2^(prime) −1=prime were true, then 2^(2^(prime) −1) −1=prime is also true, i.e. if we know a prime say p, we will know infinite other primes: 2^p −1, 2^(2^p −1) −1,2^(2^(2^p −1) −1) −1,.... but this is not the case. in fact till now the largest prime number we know is 2^(282589933) −1. if you can find the next one, your name will go into the history.

$${if}\:\mathrm{2}^{{prime}} −\mathrm{1}={prime}\:\boldsymbol{{were}}\:{true},\:{then} \\ $$$$\mathrm{2}^{\mathrm{2}^{{prime}} −\mathrm{1}} −\mathrm{1}={prime}\:{is}\:{also}\:{true},\:{i}.{e}. \\ $$$${if}\:{we}\:{know}\:{a}\:{prime}\:{say}\:{p},\:{we}\:{will}\:{know}\: \\ $$$${infinite}\:{other}\:{primes}: \\ $$$$\mathrm{2}^{{p}} −\mathrm{1},\:\mathrm{2}^{\mathrm{2}^{{p}} −\mathrm{1}} −\mathrm{1},\mathrm{2}^{\mathrm{2}^{\mathrm{2}^{{p}} −\mathrm{1}} −\mathrm{1}} −\mathrm{1},.... \\ $$$${but}\:{this}\:{is}\:{not}\:{the}\:{case}.\:{in}\:{fact}\:{till}\:{now} \\ $$$${the}\:{largest}\:{prime}\:{number}\:{we}\:{know} \\ $$$${is}\:\mathrm{2}^{\mathrm{282589933}} −\mathrm{1}.\:{if}\:{you}\:{can}\:{find}\:{the}\:{next} \\ $$$${one},\:{your}\:{name}\:{will}\:{go}\:{into}\:{the}\:{history}. \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 21/Mar/19

bah darun yhank you sir...

$${bah}\:{darun}\:{yhank}\:{you}\:{sir}... \\ $$

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