Mersenne Numbers
Published On :2021-08-30 13:37:00
Mersenne Numbers
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We;now;giving;a;method;for;finding;large;prime;numbers,;which;is;due;to;\\
Mersenne.\\
Definition:;The;numbers;of;the;form;;;;{M_{n;}} = {2^{n;}} - 1,;;n ge ;1;;are;called;Mersenne;\\
numbers,;and;those;which;are;prime;are;called;Mersenne;primes.\\
;;;It;is;clear;that;if;n = mk;is;composite,;then\\
;{2^n} - 1 = {left( {{2^m}} right)^k} - 1 = left( {{2^m} - 1} right)left( {;{{left( {{2^m}} right)}^{k - 1}} + {{left( {{2^m}} right)}^{k - 2}} + ldots + {2^m} + 1} right)is;composite.\\
In;the;preface;of;his;Cogitata;physico - mathematical;left( {1644} right),;the;French;\\
monk;Mersenne;stated;that;the;numbers;;{2^p} - 1;were;prime;for;\\
p; = ;;2;,;3;,;5;,;7;,;13;,;17;,;19;,;31;,;67;,;127;;and;257;and;composite;for;all;other;\\
primes;p; < ;257;.\\
;;;;In;left( {1772} right);Euler;verified;that;;{M_{31}};;was;prime,;but;;{M_{61;}};,;{M_{89}};,;{M_{107}};are;\\
primes;,;while;;{M_{67}};,;;{M_{257}};are;composite,and;this;contradicts;with;the;\\
declaration;of;Mersenne;in;five;points.\\
But;what;we;mentioned;earlier;does;not;mean;that;Mersenne;numbers;\\
;are;not;important,;as;many;important;theories;have;been;based;on\\
them;to;determine;the;primality.\\
We;can;use;Wolfram;Mathematica;to;find;out;the;correct;prime;numbers;less;\\
than;257;and;they;are;as;follows:p; = ;2,3,5,7,13,17,19,31,61,89,107,127.\\
Theorem;:;If;q; = ;2n; + 1;;is;prime,;then;we;have;the;following:;\\
left( a right);q;|;{M_n} + 2;,;provided;that;q; equiv ;3;left( {;mod;8;} right);or;q; equiv ;5;left( {;mod;8} right).\\
left( b right);q;|;{M_n};,;provided;that;q; equiv ;1;left( {;mod;8;} right);or;q; equiv ;7;left( {;mod;8;} right).\\
See;left( {Burton,;2011,;p.229} right).;\\
Examples:;\\
left( 1 right);Let{rm{ }}n = 11,{rm{ }}then{rm{ }}q{rm{ }} = {rm{ }}23,{rm{ }}since{rm{ }}q{rm{ }} equiv {rm{ }}7{rm{ }}left( {{rm{ }}mod{rm{ }}8} right).{rm{ }}Thus;23|{M_{11}},{rm{ }}left( {Verified{rm{ }}by{rm{ }}Mathematica} right).\\
left( 2 right){rm{ }}Let{rm{ }}n = 14,{rm{ }}then{rm{ }}q = 29,{rm{ }}since{rm{ }}q{rm{ }} equiv {rm{ }}5{rm{ }}left( {mod{rm{ }}8} right).{rm{ }}Thus{rm{ }}29|{M_{14}}{rm{ + 2}},{rm{ }}left( {Verified{rm{ }}by{rm{ }}Mathematica} right).
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