On the verification of Clark's example of a euclidean but not norm-euclidean number field.
On upper triangular nonnegative matrices
We first investigate factorizations of elements of the semigroup of upper triangular matrices with nonnegative entries and nonzero determinant, provide a formula for , and, given , also provide formulas for , and . As a consequence, open problem 2 and problem 4 presented in N. Baeth et al. (2011), are partly answered. Secondly, we study the semigroup of upper triangular matrices with only positive integral entries, compute some invariants of such semigroup, and also partly answer open Problem...
On zero-testing and interpolation of sums of characters.
Palindromic squares for various number system bases
Periodic Gaussian moats.
Polynomials taking prime values. (Polynômes prenant des valeurs premières.)
Portraits of Numbers
Power integral bases in cubic relative extensions.
Power integral bases in the family of simplest quartic fields.
Poznámka k číslům Bernoulliho
Practical Aurifeuillian factorization
We describe a simple procedure to find Aurifeuillian factors of values of cyclotomic polynomials for integers and . Assuming a suitable Riemann Hypothesis, the algorithm runs in deterministic time , using space, where .
Primality test for numbers of the form
We describe a primality test for with an odd prime p and a positive integer n, which are a particular type of generalized Fermat numbers. We also present special primality criteria for all odd prime numbers p not exceeding 19. All these primality tests run in deterministic polynomial time in the input size log₂M. A special 2pth power reciprocity law is used to deduce our result.
Primality tests using algebraic groups.
Prime and composite terms in Sloane's sequence A056542.
Prime ideal factorization in a number field via Newton polygons
Let be a number field defined by an irreducible polynomial and its ring of integers. For every prime integer , we give sufficient and necessary conditions on that guarantee the existence of exactly prime ideals of lying above , where factors into powers of monic irreducible polynomials in . The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly prime ideals of lying above . We further specify...
Prime numbers with a fixed number of one bits or zero bits in their binary representation.
Prime Pythagorean triangles.
Primes of the form .
Primitive divisors of Lucas and Lehmer sequences, II
Let and are conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all and with h, the -th element of these sequences has a primitive divisor for . In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.
Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers
Let d be a fixed positive integer. A Lucas d-pseudoprime is a Lucas pseudoprime N for which there exists a Lucas sequence U(P,Q) such that the rank of appearance of N in U(P,Q) is exactly (N-ε(N))/d, where the signature ε(N) = (D/N) is given by the Jacobi symbol with respect to the discriminant D of U. A Lucas d-pseudoprime N is a primitive Lucas d-pseudoprime if (N-ε(N))/d is the maximal rank of N among Lucas sequences U(P,Q) that exhibit N as a Lucas pseudoprime. We derive...