Gaussian Mersenne and Eisenstein Mersenne primes

Abstract

Abstract. The Biquadratic Reciprocity Law is used to produce a determin-

Efficient finite groups arising in the study of relative asphericity

“…) and so we obtain the presentation (7). This presentation shows that J ′ is a homomorphic image of the cyclically presented…”

“…The presentations (6), (7) suggest generalizing the concept of cyclically presented groups, defined at (1), to bicyclically presented groups G r (w, v) defined by the presentation Then for n = 4 or 6 the presentations (6), (7) give…”

“…It follows from Theorem C that a 6 (8, 1) = (4 4 + 3 4 ) 2 = 337 2 and a 6 (32, 1) = (4 16 +3 16 ) 2 = 4338014017 2 . It will follow from Theorem 6.2 that in fact Further background and a primality test for Gaussian-Mersenne numbers in given in [7] and a list of known primes GM m is available at [39, A182300]. As a corollary to Theorem C we note the orders of the abelianizations of certain cyclically presented groups that have been considered in the literature.…”

Primality test for mersenne numbers using elliptic curves

“…Then or depending on whether is a quadratic nonresidue or a quadratic residue modulo p. [1,2]; if(isprime(k)==0, return("is composite"); ); if((p+1)/(2^k)!=1, return("The number is not Merssene prime"); ); if(k<3, return("k is not on the condition k>=3"); ); x=3; for(i=1,k-1,z=Mod(x^3-6*x,p);g=gcd(lift(z),p); if(g>1, return("Is Composite"); ); x1=Mod(4*(x^3-6*x),p); if(x1==0, return("Is Composite"); ); x=Mod((x^4+12*x^2+36)/(x1),p); ); z=Mod(x^3-6*x,p); if(z!=0, return("Is Composite");, return("Is Prime") );}…”

Mersenne Primes in Real Quadratic Fields

“…In both the papers the concept of Mersenne primes is used to give a valid definition of even-perfect numbers. Recently Pedro Berrizbeitia and Boris Iskra studied Mersenne primes over Gaussian integers and Eisenstein integers [3]. The primality of Gaussian Mersenne numbers and Eisenstein Mersenne numbers are tested using biquadratic reciprocity and cubic reciprocity laws respectively.…”