The Biquadratic Reciprocity Law is used to produce a deterministic primality test for Gaussian Mersenne norms which is analogous to the Lucas—Lehmer test for Mersenne numbers. It is shown that the proposed test could not have been obtained from the Quadratic Reciprocity Law and Proth's Theorem. Other properties of Gaussian Mersenne norms that contribute to the search for large primes are given. The Cubic Reciprocity Law is used to produce a primality test for Eisenstein Mersenne norms. The search for primes in both families (Gaussian Mersenne and Eisenstein Mersenne norms) was implemented in 2004 and ended in November 2005, when the largest primes, known at the time in each family, were found.
This journal, begun in 1943 as Mathematical Tables and Other Aids to Computation, publishes original articles on all aspects of numerical mathematics, book reviews, mathematical tables, and technical notes. It is devoted to advances in numerical analysis, the application of computational methods, high speed calculating, and other aids to computation.
Founded in 1888, to further mathematical research and scholarship, the 30,000-member American Mathematical Society provides programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and everyday life. The headquarters of the AMS are in Providence, Rhode Island. The Society also maintains a government relations office in Washington, D.C., the Mathematical Reviews editorial office in Ann Arbor, Michigan, and a warehouse and distribution facility in Pawtucket, Rhode Island. The Society has approximately 240 employees.
This item is part of a JSTOR Collection.
For terms and use, please refer to our
Mathematics of Computation
© 2010 American Mathematical Society