## NEW POLYNOMIALS PRODUCING ABSOLUTE PSEUDOPRIMES WITH ANY NUMBER OF PRIME FACTORS (2007)

### BibTeX

@MISC{Nakamula07newpolynomials,

author = {Ken Nakamula and Hirofumi Tsumura and Hiroaki Komai},

title = {NEW POLYNOMIALS PRODUCING ABSOLUTE PSEUDOPRIMES WITH ANY NUMBER OF PRIME FACTORS},

year = {2007}

}

### OpenURL

### Abstract

Abstract. In this paper, we introduce a certain method to construct polynomials producing many absolute pseudoprimes. By this method, we give new polynomials producing absolute pseudoprimes with any fixed number of prime factors which can be viewed as a generalization of Chernick’s result. By the similar method, we give another type of polynomials producing many absolute pseudoprimes. As concrete examples, we tabulate the counts of such numbers of our forms. 1.

### Citations

166 |
An introduction to the theory of numbers, Fifth edition
- Hardy, Wright
- 1980
(Show Context)
Citation Context ...(see [4] §3). Denote by P(N) the probability of N being prime for N ∈ N. By the Prime Number Theorem, we have P(N) ∼ 1 (N → ∞). (3.1) log N On the other hand, it follows from the Mertens theorem (see =-=[7]-=- §22.8) that ∏ p − 1 2e−γ ∼ (N → ∞). (3.2) p log N p:prime p≤ √ N First, we consider U4,4(m). Let u = q·r·s·t, where q = 20m+1, r = 80m+1, s = 100m + 1 and t = 200m + 1. By the Prime Number Theorem, t... |

133 |
Some problems on partitio numerorum III : On the expression of a number as a sum of primes
- Littlewood
- 1923
(Show Context)
Citation Context ...with k prime factors. As concrete examples, we tabulate the counts of such numbers by using the method similar to Dubner’s one. Indeed, Dubner turned the method of Hardy and Littlewood precisely (see =-=[6]-=-), and tabulated the counts of absolute pseudoprimes of the form U3,3(M) (see [4]). We make use of his method, and tabulate the counts of absolute pseudoprimes of the form U4,4(M), U5,5(M) and W4(3M).... |

13 |
There are infinitely many Carmichael
- Alford, Granville, et al.
- 1994
(Show Context)
Citation Context ...e if and only if N is squarefree and p − 1 divides N − 1 for every prime p dividing N. From this, it easily follows that the number of prime divisors of the absolute pseudoprime is at least three. In =-=[1]-=-, Alford, Granville and Pomerance showed that there are infinitely many absolute pseudoprimes. Furthermore Lőw and Niebuhr [8] introduced a new algorithm for constructing absolute pseudoprimes with a ... |

12 | Two contradictory conjectures concerning Carmichael numbers
- Granville, Pomerance
(Show Context)
Citation Context ...int of what is called the k-tuple prime conjecture (see [3] Chapter 1), it seem to be natural that Uk(m) produces infinitely many kcomponent absolute pseudoprimes. Furthermore Granville and Pomerance =-=[5]-=- considered this deeply and gave the general theory about estimation of the number of k-component absolute pseudoprimes under the Hardy-Littlewood conjecture. As they mentioned in [5] Section 2, it is... |

11 |
On Fermat T s Simple Theorem
- Chernick
(Show Context)
Citation Context ...αi}, {βi}) = (αim + βi) (m ∈ N), where αi ∈ N and βi ∈ Z (1 ≤ i ≤ k) which satisfy the congruences Uk(m) ≡ 1 (mod αim + βi − 1) for any m ∈ N (1 ≤ i ≤ k). This has been already considered by Chernick =-=[2]-=-. Chernick called Uk(m) the universal form. For i=1 This research is partially supported by the JSPS Grand-in-Aid for Scientific Research No. 16340011. 12 KEN NAKAMULA, HIROFUMI TSUMURA, AND HIROAKI ... |

11 |
Prime Numbers: A
- Crandall, Pomerance
(Show Context)
Citation Context ...construct many k-component universal forms because we cannot obtain even one kcomponent absolute pseudoprime for an arbitrary k. From the viewpoint of what is called the k-tuple prime conjecture (see =-=[3]-=- Chapter 1), it seem to be natural that Uk(m) produces infinitely many kcomponent absolute pseudoprimes. Furthermore Granville and Pomerance [5] considered this deeply and gave the general theory abou... |

3 | Numbers of the form (6m+1)(12m+1)(18m+1
- Dubner
(Show Context)
Citation Context ...s by using the method similar to Dubner’s one. Indeed, Dubner turned the method of Hardy and Littlewood precisely (see [6]), and tabulated the counts of absolute pseudoprimes of the form U3,3(M) (see =-=[4]-=-). We make use of his method, and tabulate the counts of absolute pseudoprimes of the form U4,4(M), U5,5(M) and W4(3M). 2. Polynomials producing many absolute pseudoprimes First we give the following ... |

1 | A new algorithm for constructing large Carmichael
- Lőh, Niebuhr
- 1996
(Show Context)
Citation Context ...umber of prime divisors of the absolute pseudoprime is at least three. In [1], Alford, Granville and Pomerance showed that there are infinitely many absolute pseudoprimes. Furthermore Lőw and Niebuhr =-=[8]-=- introduced a new algorithm for constructing absolute pseudoprimes with a large number of prime factors. However, it is still open that there are infinitely many kcomponent absolute pseudoprimes for e... |