Mersenne Numbers

 Mersenne Numbers 

$$\begin{array}{l}Wenowgivingamethodforfindinglargeprimenumbers,whichisdueto\\ \\ Mersenne.\\ \\ Definition:Thenumbersoftheform{M}_n=2^n-1,n\ge 1arecalledMersenne\\ \\ numbers,andthosewhichareprimearecalledMersenneprimes.\\ \\ Itisclearthatifn=mkiscomposite,then\\ \\ 2^n-1={\left(2^m\right)}^k-1=\left(2^m-1\right)\left({\left(2^m\right)}^{k-1}+{\left(2^m\right)}^{k-2}+\dots +2^m+1\right)iscomposite.\\ \\ IntheprefaceofhisCogitataphysico-mathematical\left(1644\right),theFrench\\ \\ monkMersennestatedthatthenumbers{2}^p-1wereprimefor\\ \\ p=2,3,5,7,13,17,19,31,67,127and257andcompositeforallother\\ \\ primesp<257.\\ \\ In\left(1772\right)Eulerverifiedthat{M}_{31}wasprime,but{M}_{61},M_{89},M_{107}are\\ \\ primes,while{M}_{67},M_{257}arecomposite,andthiscontradictswiththe\\ \\ declarationofMersenneinfivepoints.\\ \\ ButwhatwementionedearlierdoesnotmeanthatMersennenumbers\\ \\ arenotimportant,asmanyimportanttheorieshavebeenbasedon\\ \\ themtodeterminetheprimality.\\ \\ WecanuseWolframMathematicatofindoutthecorrectprimenumbersless\\ \\ than257andtheyareasfollows:p=2,3,5,7,13,17,19,31,61,89,107,127.\\ \\ Theorem:Ifq=2n+1isprime,thenwehavethefollowing:\\ \\ \left(a\right)q|M_n+2,providedthatq\equiv 3\left(mod8\right)orq\equiv 5\left(mod8\right).\\ \\ \left(b\right)q|M_n,providedthatq\equiv 1\left(mod8\right)orq\equiv 7\left(mod8\right).\\ \\ See\left(Burton,2011,p.229\right).\\ \\ Examples:\\ \\ \left(1\right)Letn=11,thenq=23,sinceq\equiv 7\left(mod8\right).Thus23|M_{11},\left(VerifiedbyMathematica\right).\\ \\ \left(2\right)Letn=14,thenq=29,sinceq\equiv 5\left(mod8\right).Thus29|M_{14}+2,\left(VerifiedbyMathematica\right).\end{array}$$