A new algorithm for constructing large Carmichael numbers

We describe an algorithm for constructing Carmichael numbers with a large number of prime factors . This algorithm starts with a given number , representing the value of the Carmichael function . We found Carmichael numbers with up to factors.

     

Density of Carmichael numbers with three prime factors

We get an upper bound of on the number of Carmichael numbers with exactly three prime factors.

     

广义Carmichael数

设n是一个合数,Z_n表示模n的剩余类环,r(x)∈Z_n[x]是一个首一的k(>0)次不可约多项式。本文引入n是k阶摸r(x)的Carmichael数的定义,全体这样的数记为集C_(k,r)(x),由此给出k阶Carmichael数集:C_k={∪C_(k,r)(x)|r(x)过全体Z_n上的首一k次不可约多项式}。显然C_1表示通常的Carmichael数集。作者得到了n∈C_(k,r(x))的一个充要条件,进而得到n∈C_k的一个充要条件及n∈C_2的一个更易计算的充要条件,还证明了C_1(?)C_2以及|C_2|=∞。     

Square-free values of the Carmichael function

We obtain an asymptotic formula for the number of square-free values among p−1, for primes p?x, and we apply it to derive the following asymptotic formula for L(x), the number of square-free values of the Carmichael function λ(n) for 1?n?x,

     

关于3阶Carmichael数

朱文余和孙琦（见《数学进展》,2004,33（4）：505-507）提出了关于3阶Carmichael数的三个问题,我们（见《四川大学学报（自然科学版）》,2006,43（6）：1197-1201）肯定地回答了问题1．本文模仿Howe的寻找严格2阶Carmichael数（见Mathematics of Computation,2000,69（232）：1711—1719）的方法,提出寻找满足某种条件的3阶Carmichael数的方法,并用这种方法确实找到了几百个这样的数,因而完全肯定地回答了问题2．     

Carmichael numbers in number rings

We generalize Carmichael numbers to ideals in number rings and prove a generalization of Korselt's Criterion for these Carmichael ideals. We investigate when Carmichael numbers in the integers generate Carmichael ideals in the algebraic integers of abelian number fields. In particular, we show that given any composite integer n, there exist infinitely many quadratic number fields in which n is not Carmichael. Finally, we show that there are infinitely many abelian number fields K with discriminant relatively prime to n such that n is not Carmichael in K.     

Two contradictory conjectures concerning Carmichael numbers

Erdos conjectured that there are Carmichael numbers up to , whereas Shanks was skeptical as to whether one might even find an up to which there are more than Carmichael numbers. Alford, Granville and Pomerance showed that there are more than Carmichael numbers up to , and gave arguments which even convinced Shanks (in person-to-person discussions) that Erdos must be correct. Nonetheless, Shanks's skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch's extended new data), and so we herein derive conjectures that are consistent with Shanks's observations, while fitting in with the viewpoint of Erdos and the results of Alford, Granville and Pomerance.

     

On Kelley's intersection numbers

We introduce a notion of weak intersection number of a collection of sets, modifying the notion of intersection number due to J.L. Kelley, and obtain an analogue of Kelley's characterization of Boolean algebras which support a finitely additive strictly positive measure. We also consider graph-theoretic reformulations of the notions of intersection number and weak intersection number.

     

On Fibonacci numbers which are elliptic Carmichael

Here, we show that if E is a CM elliptic curve with CM field different from \({\mathbb {Q}}\left( {\sqrt{-1}}\right) \), then the set of n for which the nth Fibonacci number \(F_n\) is elliptic Carmichael for E is of asymptotic density zero.     

Thomason's theorem for varieties over algebraically closed fields

We present a novel proof of Thomason's theorem relating Bott inverted algebraic -theory with finite coefficients and étale cohomology for smooth varieties over algebraically closed ground fields. Our proof involves first introducing a new theory, which we term algebraic -homology, and proving it satisfies étale descent (with finite coefficients) on the category of normal, Cohen-Macaulay varieties. Then, we prove algebraic -homology and algebraic -theory (each taken with finite coefficients) coincide on smooth varieties upon inverting the Bott element.

     

Integral crystalline cohomology over very ramified valuation rings

We explain how to set up an integral version ( as opposed to ) of Fontaine's comparison between crystalline and étale cohomology, over -adic fields with arbitrary ramification index. The main results then are that Fontaine's map respects integrality of Tate-cycles, and a construction of versal deformations of -divisible groups with Tate-cycles. An appendix deals with finite generation of crystalline cohomology.

     

The clique numbers of regular graphs of matrix algebras are finite

Let F be a field, char(F)≠2, and SGL_n(F), where n is a positive integer. In this paper we show that if for every distinct elements x,yS, x+y is singular, then S is finite. We conjecture that this result is true if one replaces field with a division ring.     

Young-Fibonacci insertion, tableauhedron and Kostka numbers

This work is first concerned with some properties of the Young-Fibonacci insertion algorithm and its relation with Fomin's growth diagrams. It also investigates a relation between the combinatorics of Young-Fibonacci tableaux and the study of Okada's algebras associated to the Young-Fibonacci lattice. The original algorithm was introduced by Roby and we redefine it in such a way that both the insertion and recording tableaux of any permutation are conveniently interpreted as saturated chains in the Young-Fibonacci lattice. Using our conventions, we give a simpler proof of a property of Killpatrick's evacuation algorithm for Fibonacci tableaux. It also appears that this evacuation is no longer needed in making Roby's and Fomin's constructions coincide. We provide the set of Young-Fibonacci tableaux of size n with a structure of graded poset called tableauhedron, induced by the weak order of the symmetric group, and realized by transitive closure of elementary transformations on tableaux. We show that this poset gives a combinatorial interpretation of the coefficients of the transition matrix from the analogue of complete symmetric functions to analogue of the Schur functions in Okada's algebra associated to the Young-Fibonacci lattice. We prove a similar result relating usual Kostka numbers with four partial orders on Young tableaux, studied by Melnikov and Taskin.     

Square-free Lucas <Emphasis Type="Italic">d</Emphasis>-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 N in U(P, Q) is exactly (N − ε(N))/d, where ε is the signature of U(P, Q). We prove here that all but a finite number of Lucas d-pseudoprimes are square free. We also prove that all but a finite number of Lucas d-pseudoprimes are Carmichael-Lucas numbers.     

Genus expanded cut-and-join operators and generalized Hurwtiz numbers

To distinguish the contributions to the generalized Hurwitz number of the source Riemann surface with different genus, by observing carefully the symplectic surgery and the gluing formulas of the relative GW-invariants, we define the genus expanded cut-and-join operators. Moreover all normalized the genus expanded cut-and-join operators with same degree form a differential algebra, which is isomorphic to the central subalgebra of the symmetric group algebra. As an application, we get some differential equations for the generating functions of the generalized Hurwitz numbers for the source Riemann surface with different genus, thus we can express the generating functions in terms of the genus expanded cut-and-join operators.     

Betti numbers for p -stable ideals

Abstract

All Riemannian algebras of dimension ≤3 are classified in this paper.     

Set-theoretic complete intersections in characteristic

We describe a class of toric varieties which are set-theoretic complete intersections only over fields of one positive characteristic .

     

Multichromatic numbers,star chromatic numbers and Kneser graphs

We investigate the relation between the multichromatic number (discussed by Stahl and by Hilton, Rado and Scott) and the star chromatic number (introduced by Vince) of a graph. Denoting these by χ* and η*, the work of the above authors shows that χ*(G) = η*(G) if G is bipartite, an odd cycle or a complete graph. We show that χ*(G) ≤ η*(G) for any finite simple graph G. We consider the Kneser graphs , for which χ* = m/n and η*(G)/χ*(G) is unbounded above. We investigate particular classes of these graphs and show that η* = 3 and η* = 4; (n ≥ 1), and η* = m - 2; (m ≥ 4). © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 137–145, 1997