Remark on factorials that are products of factorials

In a paper published in 1993, Erdös proved that if n! = a! b!, where 1 < a ≤ b, then the difference between n and b does not exceed 5 log log n for large enough n. In the present paper, we improve this upper bound to ((1 + ε)/ log 2) log log n and generalize it to the equation a ₁!a ₂! ... a _k ! = n!. In a recent paper, F. Luca proved that n ? b = 1 for large enough n provided that the ABC-hypothesis holds.     

Factorization of factorials

If factorials enter into product-quotient expressions, not allowing approximative calculation, the factorization into powers of primes is recommended. The paper describes a computer program for this process, based on a known formula, which should be applied differently, according as the primes are large or small. A discussion shows how to distinguish between large and small in order to minimize the computing time.     

Remarks on a sum containing factorials

It is shown how readily a sum containing factorials, which was considered recently by Samoletov (J. Comput. Appl. Math. 131 (2001) 503–504), would follow from substantially more general known results.     

On the complexity of calculating factorials

It is shown that n! can be evaluated with time complexity O(log log nM (n log n)), where M(n) is the complexity of multiplying two n-digit numbers together. This is effected, in part, by writing n! in terms of its prime factors. In conjunction with a fast multiplication this yields an O(n(log n log log n)²) complexity algorithm for n!. This might be compared to computing n! by multiplying 1 times 2 times 3, etc., which is ω(n² log n) and also to computing n! by binary splitting which is O(log nM(n log n)).     

Asymptotic formulas for g-gonal sequence factorials

In this note, we provide basic asymptotic formulas for approximating large g-gonal sequence factorials by using Stirling and Burnside asymptotic approximation formulas for large factorials. More accurate asymptotic approximation formulas for large g-gonal sequence factorials resulting from some recent, more accurate asymptotic formulas for large factorials that have appeared in the literature are presented.     

Some properties of infinite factorials

With the Axiom of Choice , for any infinite cardinal but, without , we cannot conclude any relationship between and for an arbitrary infinite cardinal . In this paper, we give some properties of in the absence of and compare them to those of for an infinite cardinal . Among our results, we show that “ for any infinite cardinal and any natural number n” is provable in although “ for any infinite cardinal ” is not.     

On non-orthogonal main effect plans for asymmetrical factorials

Summary Following the lines of Raktoe and Federer [19] a unified approach for constructing main effect plans in any factorials wherek _i's are the numbers of equispaced levels of each of then _i factors, andk _i's are not necessarily primes or prime powers and need not satisty any relations among themselves, is presented. The method consists of, first, dividing the totality of treatment combinations, omitting, of course, some, if necessary, in to pairs such that the differences within the pairs are clear of ‘even’ effects and the sums are clear of ‘odd’ effects, and then, depending on the number of error d.f. wanted, selecting a suitable sub-set of these pairs which lead to the solution of the estimates of main effects. A general class of non-orthogonal main effect plans for 2 ^m ×2 ⁿ factorials is proposed. Information matrices and their inverses for such plans are worked out. An example followed by discussions and comparison statements is presented.     

The exponents in the prime decomposition of factorials

Let ν_p(n) be the exponent of p in the prime decomposition of n. We show that for different primes p, q satisfying some mild constraints the integers ν_p(n!) and ν_q(n!) cannot both be of a rather special form.     

Uniformity pattern and related criteria for two-level factorials

In this paper,the study of projection properties of two-level factorials in view ofgeometry is reported.The concept of uniformity pattern is defined.Based on this new con-cept,criteria of uniformity resolution and minimum projection uniformity are proposed forcomparing two-level factorials.Relationship between minimum projection uniformity andother criteria such as minimum aberration,generalized minimum aberration and orthogo-nality is made explict.This close relationship raises the hope of improving the connectionbetween uniform design theory and factorial design theory.Our results provide a justifi-cation of orthogonality,minimum aberration,and generalized minimum aberration from anatural geometrical interpretation.     

A note on optimal foldover four-level factorials

The foldover is a quick and useful technique in construction of fractional factorial designs, which typically releases aliased factors or interactions. The issue of employing the uniformity criterion measured by the centered L ₂-discrepancy to assess the optimal foldover plans was studied for four-level design. A new analytical expression and a new lower bound of the centered L ₂-discrepancy for fourlevel combined design under a general foldover plan are respectively obtained. A necessary condition for the existence of an optimal foldover plan meeting this lower bound was described. An algorithm for searching the optimal four-level foldover plans is also developed. Illustrative examples are provided, where numerical studies lend further support to our theoretical results. These results may help to provide some powerful and efficient algorithms for searching the optimal four-level foldover plans.     

Waring type congruences involving factorials modulo a prime

In this paper we prove that any residue class λ modulo a large prime number p can be represented in the form

Lee discrepancy on asymmetrical factorials with two- and three-levels

Lee discrepancy has been employed to measure the uniformity of fractional factorials.In this paper,we further study the statistical justification of Lee discrepancy on asymmetrical factorials.We will give an expression of the Lee discrepancy of asymmetrical factorials with two-and three-levels in terms of quadric form,present a connection between Lee discrepancy,orthogonality and minimum moment aberration,and obtain a lower bound of Lee discrepancy of asymmetrical factorials with two-and three-levels.     

Application of minimum projection uniformity criterion in complementary designs for q-level factorials

We study the complementary design problem, which is to express the uniformity pattern of a q-level design in terms of that of its complementary design. Here, a pair of complementary designs form a design in which all the Hamming distances of any two distinct runs are the same, and the uniformity pattern proposed by H. Qin, Z. Wang, and K. Chatterjee [J. Statist. Plann. Inference, 2012, 142: 1170–1177] comes from discrete discrepancy for q-level designs. Based on relationships of the uniformity pattern between a pair of complementary designs, we propose a minimum projection uniformity rule to assess and compare q-level factorials.     

Divided differences and a unified approach to evaluating determinants involving generalized factorials

By employing divided differences, a unified approach to the evaluation of some determinant involving generalized factorials is proposed. Previous generalizations of the Vandermonde determinant and the Cauchy determinant due to Chu-Claudio, Chu-Wang-Zhang and Johnson are included as special cases of our unified treatment.     

2-Adic valuations of certain ratios of products of factorials and applications

We prove the conjecture of Falikman-Friedland-Loewy on the parity of the degrees of projective varieties of n×n complex symmetric matrices of rank at most k. We also characterize the parity of the degrees of projective varieties of n×n complex skew symmetric matrices of rank at most 2p. We give recursive relations which determine the parity of the degrees of projective varieties of m×n complex matrices of rank at most k. In the case the degrees of these varieties are odd, we characterize the minimal dimensions of subspaces of n×n skew symmetric real matrices and of m×n real matrices containing a nonzero matrix of rank at most k. The parity questions studied here are also of combinatorial interest since they concern the parity of the number of plane partitions contained in a given box, on the one hand, and the parity of the number of symplectic tableaux of rectangular shape, on the other hand.     

The order of appearance of factorials in the Fibonacci sequence and certain Diophantine equations

Periodica Mathematica Hungarica - Let $$F_n$$ be the nth Fibonacci number. The order (or the rank) of appearance of m in the Fibonacci sequence, denoted by z(m), is the smallest positive integer k...     

Search designs for searching for one among the two-and three-factor interaction effects in the general symmetric and asymmetric factorials

This paper describes the construction of search designs which permit the estimation of the general mean and main-effects, and allow the search for and estimation of one possibly unknown non-zero effect among the two-and three-factor interactions in the general symmetric and asymmetric factorial set-up.