pq-Catalan numbers and squarefree binomial coefficients

AbstractIn this paper, we consider the generalized Catalan numbers F(s,n)=1(s−1)n+1(snn), which we call s-Catalan numbers. For p prime, we find all positive integers n such that pq divides F(pq,n), and also determine all distinct residues of F(pq,n)(modpq), q⩾1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. In the second part of the paper we prove that if pq⩽99999, then (pqn+1n) is not squarefree for n⩾τ1(pq) sufficiently large (τ1(pq) computable). Moreover, using the results of the first part, we find n<τ1(pq) (in base p), for which (pqn+1n) may be squarefree. As consequences, we obtain that (4n+1n) is squarefree only for n=1,3,45, and (9n+1n) is squarefree only for n=1,4,10

The Euler function of Fibonacci and Lucas numbers and factorials

Here, we look at the Fibonacci and Lucas numbers whose Euler function is a factorial, as well as Lucas numbers whose Euler function is a product of power of two and power of thre

Rotation symmetric Boolean functions---count and cryptographic properties

The article of record as published may be located at http://dx.doi.org/10.1.1.137.6388Rotation symmetric (RotS) Boolean functions have been used as components of different cryptosystems. This class of Boolean functions are invariant under circular translation of indices. Using Burnsideﾒs lemma it can be seen that the number of n-variable rotation symmetric Boolean functions is 2gn, where gn = 1 nPt|n (t) 2n t , and (.) is the Euler phi-function. In this paper, we find the number of short and long cycles of elements in Fn2 having fixed weight, under the RotS action. As a consequence we obtain the number of homogeneous RotS functions having algebraic degree w. Our results make the search space of RotS functions much reduced and we successfully analyzed important cryptographic properties of such functions by executing computer programs. We study RotS bent functions up to 10 variables and observe (experimentally) that there is no homogeneous rotation symmetric bent function having degree > 2. Further, we studied the RotS functions on 5, 6, 7 variables by computer search for correlation immunity and propagation characteristics and found some functions with very good cryptographic properties which were not known earlier

F_1 F_2 F_3 F_4 F_5 F_6 F_8 F_10 F_12=11!

In this paper, we show that the equality appearing in the title gives the largest solutioin of the diopahntine equation where Fn1...Fnk = M1!...mt!, where 0 <n1 <... <nk and 1< m1 < m2<...<mt are integers

Aliquots sums of Fibonacci numbers

Here, we investigate the Fibonacci numbers whose sum of aliquot divisors is also a Fibonacci number (the prime Fibonacci numbers have this propert

Recurrence relations for a third-order family of methods in Banach Spaces

Recently, PArida and Gupta (J. Comp. Appl.Math. 2-6 (2007), 873-877) used Rall's recurrence relations approach (from 1961) to approximate roots of nonlinear equations, by developing several methods, the latest of which is free of second derivative and it is of third order. In this paper, we use an idea of Kou and Li (appl. Math. Comp. 187 (2007), 1027-1032) and modify the approach of Parida and Gupta, obtaining yet another third-order method to approximate a solution of a non-linear equation in a Banach space. We give several applications to our method

The c-differential uniformity and boomerang uniformity of two classes of permutation polynomials

The article of record as published may be found at http://dx.doi.org/10.1109/TIT.2021.3123104The Difference Distribution Table (DDT) and the differential uniformity play a major role for the design of substitution boxes in block ciphers, since they indicate the func- tion’s resistance against differential cryptanalysis. This concept was extended recently to c-DDT and c-differential uniformity, which have the potential of extending differential cryptanalysis. Recently, a new theoretical tool, the Boomerang Connectivity Table (BCT) and the corresponding boomerang uniformity were introduced to quantify the resistance of a block cipher against boomerang-style attacks. Here we concentrate on two classes (introduced recently) of permutation polynomials over finite fields of even characteristic. For one of these, which is an involution used to construct a 4-uniform permutation, we explicitly determine the c-DDT entries and BCT entries. For the second type of function, which is a differentially 4-uniform function, we give bounds for its c-differential and boomerang uniformities.The research of Sartaj Ul Hasan is partially supported by MATRICS grant MTR/2019/000744 from the Science and Engineering Research Board, Government of India. Pantelimon Stănică acknowledges the sabbatical support from Naval Postgraduate School from September 2020 to July 2021

Asymptotic Behavior of Gaps Between Roots of Weighted Factorials

This paper is part of an undergraduate student summer project of the first author at the Naval Postgraduate School under the supervision of the second author

Bisecting binomial coefficients

In this paper, we deal with the problem of bisecting binomial coefficients. We find many (previously unknown) infinite classes of integers which admit nontrivial bisections, and a class with only trivial bisections. As a byproduct of this last construction, we show conjectures Q2 and Q4 of Cusick and Li [7]. We next find several bounds for the number of nontrivial bisections and further compute (using a supercomputer) the exact number of such bisections for n ≤ 51

On the behavior of some APN permutations under swapping points

The article of record as published may be found at https://doi.org/10.1007/s12095-021-00520-zWe define the pAPN-spectrum (which is a measure of how close a function is to being APN) of an (n,n)-function F and investigate how its size changes when two of the outputs of a given function F are swapped. We completely characterize the behavior of the pAPN-spectrum under swapping outputs when F is the inverse function over F2n . We further theoretically investigate this behavior for functions from the Gold and Welch monomial APN families, and experimentally determine the size of the pAPN-spectrum after swapping outputs for representatives from all infinite monomial APN families up to dimension n = 10; based on our computation results, we conjecture that the inverse function is the only monomial APN function for which swapping two its outputs can leave an empty pAPN-spectrum