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We describe a simple procedure to find Aurifeuillian factors of values of cyclotomic polynomials $Φ_{d} (a)$ for integers $a$ and $d > 0$ . Assuming a suitable Riemann Hypothesis, the algorithm runs in deterministic time $\tilde{O} (d^{2} L)$ , using $O (d L)$ space, where $L ≔ log (|a| + 1)$ .

Primality test for numbers of the form ${(2 p)}^{2^{n}} + 1$

Yingpu Deng, Dandan Huang (2015)

Acta Arithmetica

We describe a primality test for $M = {(2 p)}^{2^{n}} + 1$ 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.

Kida, Masanari (2004)

Experimental Mathematics

Prime and composite terms in Sloane's sequence A056542.

Müller, Tom (2005)

Journal of Integer Sequences [electronic only]

Prime ideal factorization in a number field via Newton polygons

Lhoussain El Fadil (2021)

Czechoslovak Mathematical Journal

Let $K$ be a number field defined by an irreducible polynomial $F (X) \in ℤ [X]$ and $ℤ_{K}$ its ring of integers. For every prime integer $p$ , we give sufficient and necessary conditions on $F (X)$ that guarantee the existence of exactly $r$ prime ideals of $ℤ_{K}$ lying above $p$ , where $\bar{F} (X)$ factors into powers of $r$ monic irreducible polynomials in $𝔽_{p} [X]$ . The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly $r$ prime ideals of $ℤ_{K}$ lying above $p$ . We further specify...

Prime numbers with a fixed number of one bits or zero bits in their binary representation.

Wagstaff, Samuel S.jun. (2001)

Experimental Mathematics

Prime Pythagorean triangles.

Dubner, Harvey, Forbes, Tony (2001)

Journal of Integer Sequences [electronic only]

Primes of the form $(b^{n} + 1) / (b + 1)$ .

Dubner, Harvey, Granlund, Torbjörn (2000)

Journal of Integer Sequences [electronic only]

Primitive divisors of Lucas and Lehmer sequences, II

Paul M. Voutier (1996)

Journal de théorie des nombres de Bordeaux

Let $α$ and $β$ 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 $α$ and $β$ with h $(β / α) \leq 4$ , the $n$ -th element of these sequences has a primitive divisor for $n > 30$ . 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

Walter Carlip, Lawrence Somer (2007)

Colloquium Mathematicae

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...

Products and quotients of numbers with small partial quotients

Stephen Astels (2002)

Journal de théorie des nombres de Bordeaux

For any positive integer $m$ let $F (m)$ denote the set of numbers with all partial quotients (except possibly the first) not exceeding $m$ . In this paper we characterize most products and quotients of sets of the form $F (m)$ .

Properties of some functions connected to prime numbers.

Mincu, Gabriel, Panaitopol, Laurentiu (2008)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

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