Deterministic primality test for numbers of the form odd

We use a result of E. Lehmer in cubic residuacity to find an algorithm to determine primality of numbers of the form , odd, . The algorithm represents an improvement over the more general algorithm that determines primality of numbers of the form , , presented by Berrizbeitia and Berry (1999).

     

On a class of primality criteria

Translated from Matematicheskie Zametki, Vol. 56, No. 1, pp. 146–148, July, 1994.     

Cubic reciprocity and generalised Lucas-Lehmer tests for primality of

Cubic reciprocity is used to derive primality tests analogous to the Lucas-Lehmer test for integers of the form . The test for is a minor improvement on a test derived by Williams by other means; the test for seems to be new.

     

Elliptic periods and primality proving

We construct extension rings with fast arithmetic using isogenies between elliptic curves. As an application, we give an elliptic version of the AKS primality criterion.     

Height bounds,nullstellensatz and primality

In this study, we find height bounds in the polynomial ring over the field of algebraic numbers to test the primality of an ideal. We also obtain height bounds in the arithmetic Nullstellensatz. We apply nonstandard analysis and hence our constants will be ineffective.     

Probabilistic algorithm for testing primality

We present a practical probabilistic algorithm for testing large numbers of arbitrary form for primality. The algorithm has the feature that when it determines a number composite then the result is always true, but when it asserts that a number is prime there is a provably small probability of error. The algorithm was used to generate large numbers asserted to be primes of arbitrary and special forms, including very large numbers asserted to be twin primes. Theoretical foundations as well as details of implementation and experimental results are given.     

On the primality of

For each prime , let be the product of the primes less than or equal to . We have greatly extended the range for which the primality of and are known and have found two new primes of the first form ( ) and one of the second (). We supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found.

     

Proving primality in essentially quartic random time

This paper presents an algorithm that, given a prime , finds and verifies a proof of the primality of in random time . Several practical speedups are incorporated into the algorithm and discussed in detail.

     

On the effectiveness of a generalization of Miller’s primality theorem

Berrizbeitia and Olivieri showed in a recent paper that, for any integer r$r$, the notion of ω$\omega$-prime to base a$a$ leads to a primality test for numbers n≡1$n\equiv 1$ mod r$r$, that under the Extended Riemann Hypothesis (ERH) runs in polynomial time. They showed that the complexity of their test is at most the complexity of the Miller primality test (MPT), which is O((logn)^4+o(1))$O\left({(logn)}^{4+o\left(1\right)}\right)$. They conjectured that their test is more effective than the MPT if r$r$ is large.     

Elliptic curve primality tests for Fermat and related primes

We use elliptic curves with complex multiplication to develop primality tests for Fermat primes and for primes of the form ?²³−²?−1³+1 and ?²²−²?−1²+1.     

Abstract Plancherel theorems and a Frobenius reciprocity theorem

A method for obtaining Plancherel theorems for unitary representations of Lie groups via C^∞ vector techniques is studied. The results are used to prove the nonunimodular Plancherel theorem of Moore and to study its convergence. A C^∞ Frobenius reciprocity theorem which generalizes Gelfand's duality theorem is also proven.     

Divisibity,iterated digit sums,primality tests

The divisibility of numbers is obtained by iteration of the weighted sum of their integer digits. Then evaluation of the related congruences yields information about the primality of numbers in certain recursive sequences. From the row elements in generalized Delannoy triangles, we can verify the primality of any constellation of numbers. When a number set is not a prime constellation, we can identify factors of their composite numbers. The constellation primality test is proven in all generality, and examples are given for twin primes, prime triplets, and Sophie Germain primes.      

Selmer groups and quadratic reciprocity

In this article we study the 2-Selmer groups of number fieldsF as well as some related groups, and present connections to the quadratic reciprocity law inF.