Question Number 56660 by tanmay.chaudhury50@gmail.com last updated on 20/Mar/19
Commented by kaivan.ahmadi last updated on 20/Mar/19
$${can}\:{you}\:{proof}\:{it}\:{sir}? \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 21/Mar/19
$${i}\:{think}\:{to}\:{prove} \\ $$$$\mathrm{2}^{{prime}\:{number}} \:−\mathrm{1}={prime}\:{number} \\ $$$${we}\:{need}\:{f}\left({n}\right)\:{which}\:{is}\:{prime} \\ $$$$\mathrm{2}^{{f}\left({n}\right)} −\mathrm{1}={prime}\:{number} \\ $$$${to}\:{my}\:{knowldge}\:{no}\:{such}\:{general}\:{formula}\:{f}\left({n}\right) \\ $$$${derived}\:{till}\:{now}\:{which}\:{describe}\:{prime}\:{number}. \\ $$$$ \\ $$
Commented by mr W last updated on 21/Mar/19
$${next}\:{one}: \\ $$$$\mathrm{23}\:{is}\:{prime} \\ $$$$\mathrm{2}^{\mathrm{23}} −\mathrm{1}=\mathrm{8388207}=\mathrm{47}×\mathrm{178481}\:\Rightarrow{no}\:{prime} \\ $$$${next}\:{one}: \\ $$$$\mathrm{29}\:{is}\:{prime} \\ $$$$\mathrm{2}^{\mathrm{29}} −\mathrm{1}=\mathrm{536870911}=\mathrm{233}×\mathrm{1103}×\mathrm{2089}\Rightarrow{no}\:{prime} \\ $$$${therefore} \\ $$$$\mathrm{2}^{{prime}} −\mathrm{1}\:\:\underset{{but}\:{not}\:{always}} {\overset{{maybe}} {=}}\:\:\:\:{prime} \\ $$
Commented by mr W last updated on 21/Mar/19
$${if}\:\mathrm{2}^{{prime}} −\mathrm{1}={prime}\:\boldsymbol{{were}}\:{true},\:{then} \\ $$$$\mathrm{2}^{\mathrm{2}^{{prime}} −\mathrm{1}} −\mathrm{1}={prime}\:{is}\:{also}\:{true},\:{i}.{e}. \\ $$$${if}\:{we}\:{know}\:{a}\:{prime}\:{say}\:{p},\:{we}\:{will}\:{know}\: \\ $$$${infinite}\:{other}\:{primes}: \\ $$$$\mathrm{2}^{{p}} −\mathrm{1},\:\mathrm{2}^{\mathrm{2}^{{p}} −\mathrm{1}} −\mathrm{1},\mathrm{2}^{\mathrm{2}^{\mathrm{2}^{{p}} −\mathrm{1}} −\mathrm{1}} −\mathrm{1},…. \\ $$$${but}\:{this}\:{is}\:{not}\:{the}\:{case}.\:{in}\:{fact}\:{till}\:{now} \\ $$$${the}\:{largest}\:{prime}\:{number}\:{we}\:{know} \\ $$$${is}\:\mathrm{2}^{\mathrm{282589933}} −\mathrm{1}.\:{if}\:{you}\:{can}\:{find}\:{the}\:{next} \\ $$$${one},\:{your}\:{name}\:{will}\:{go}\:{into}\:{the}\:{history}. \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 21/Mar/19
$${bah}\:{darun}\:{yhank}\:{you}\:{sir}… \\ $$