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Shaofang Hong

Publications and source records attributed to Shaofang Hong.

50 records · Page 3Linked to original sources
arXiv2013

On the periodicity of a class of arithmetic functions associated with multiplicative functions

Let $k\ge 1,a\ge 1,b\ge 0$ and $ c\ge 1$ be integers. Let $f$ be a multiplicative function with $f(n)\ne 0$ for all positive integers $n$. We define the arithmetic function $g_{k,f}$ for any positive integer $n$ by $g_{k,f}(n):=\frac{\prod_{i=0}^k f(b+a(n+ic))} {f({\rm lcm}_{0\le i\le k} \{b+a(n+ic)\})}$. We first show that $g_{k,f}$ is periodic and $c {\rm lcm}(1,...,k)$ is its period. Consequently, we provide a detailed local analysis to the periodic function $g_{k,φ}$, and determine the smallest period of $g_{k,φ}$, where $φ$ is the Euler phi function.

math.NT↗
arXiv2012

Nonlinearity of quartic rotation symmetric Boolean functions

Nonlinearity of rotation symmetric Boolean functions is an important topic on cryptography algorithm. Let $e\ge 1$ be any given integer. In this paper, we investigate the following question: Is the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ equal to its weight? We introduce some new simple sub-functions and develop new technique to get several recursive formulas. Then we use these recursive formulas to show that the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ is the same as its weight. So we answer the above question affirmatively. Finally, we conjecture that if $l\ge 4$ is an integer, then the nonlinearity of the rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}...x_{le}$ equals its weight.

cs.IT↗
arXiv2012

New Lower Bounds for the Least Common Multiples of Arithmetic Progressions

For relatively prime positive integers $u_0$ and $r$ and for $0\le k\le n$, define $u_k:=u_0+kr$. Let $L_n:={\rm lcm}(u_0, u_1, ..., u_n)$ and let $a, l\ge 2$ be any integers. In this paper, we show that, for integers $α\geq a$ and $r\geq \max(a, l-1)$ and $n\geq lαr$, we have $$L_n\geq u_0r^{(l-1)α+a-l}(r+1)^n.$$ Particularly, letting $l=3$ yields an improvement to the best previous lower bound on $L_n$ obtained by Hong and Kominers.

math.NT↗
arXiv2012

The universal Kummer congruences

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$. Using these we then establish the universal Kummer congruences modulo powers of a prime $p$ for the divided universal Bernoulli numbers ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$.

math.NT↗
arXiv2012

On deep holes of standard Reed-Solomon codes

Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the Reed-Solomon code. For the standard Reed-Solomon codes $[p-1, k]_p$ with $p$ a prime, Cheng and Murray conjectured in 2007 that there is no other deep holes except the trivial ones. In this paper, we show that this conjecture is not true. In fact, we find a new class of deep holes for standard Reed-Solomon codes $[q-1, k]_q$ with $q$ a prime power of $p$. Let $q \geq 4$ and $2 \leq k\leq q-2$. We show that the received word $u$ is a deep hole if its Lagrange interpolation polynomial is the sum of monomial of degree $q-2$ and a polynomial of degree at most $k-1$. So there are at least $2(q-1)q^k$ deep holes if $k \leq q-3$.

math.NT↗
arXiv2011

The 2-adic valuations of Stirling numbers of the second kind

In this paper, we investigate the 2-adic valuations of the Stirling numbers $S(n, k)$ of the second kind. We show that $v_2(S(4i, 5))=v_2(S(4i+3, 5))$ if and only if $i\not\equiv 7\pmod {32}$. This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for any positive integer $n$, where $s_2(n)$ is the sum of binary digits of $n$. It proves another conjecture of Amdeberhan, Manna and Moll.

math.NT↗
arXiv2011

On the integrality of the elementary symmetric functions of $1, 1/3, ..., 1/(2n-1)$

Erdos and Niven proved that for any positive integers $m$ and $d$, there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/m,1/(m+d), ..., 1/(m+nd)$ are integers. Recently, Chen and Tang proved that if $n\ge 4$, then none of the elementary symmetric functions of $1,1/2, ..., 1/n$ is an integer. In this paper, we show that if $n\ge 2$, then none of the elementary symmetric functions of $1, 1/3, ..., 1/(2n-1)$ is an integer.

math.NT↗
arXiv2011

Asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod r)

In this paper we present systematically analysis on the smallest eigenvalue of matrices associated with completely even functions (mod $r$). We obtain several theorems on the asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod $r$). In particular, we get information on the asymptotic behavior of the smallest eigenvalue of the famous Smith matrices. Finally some examples are given to demonstrate the main results.

math.NT↗
arXiv2010

The least common multiple of consecutive arithmetic progression terms

Let $k\ge 0,a\ge 1$ and $b\ge 0$ be integers. We define the arithmetic function $g_{k,a,b}$ for any positive integer $n$ by $g_{k,a,b}(n):=\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\rm lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$ Letting $a=1$ and $b=0$, then $g_{k,a,b}$ becomes the arithmetic function introduced previously by Farhi. Farhi proved that $g_{k,1,0}$ is periodic and that $k!$ is a period. Hong and Yang improved Farhi's period $k!$ to ${\rm lcm}(1,2,...,k)$ and conjectured that $\frac{{\rm lcm}(1,2,...,k,k+1)}{k+1}$ divides the smallest period of $g_{k,1,0}$. Recently, Farhi and Kane proved this conjecture and determined the smallest period of $g_{k,1,0}$. For the general integers $a\ge 1$ and $b\ge 0$, it is natural to ask the interesting question: Is $g_{k,a,b}$ periodic? If so, then what is the smallest period of $g_{k,a,b}$? We first show that the arithmetic function $g_{k,a,b}$ is periodic. Subsequently, we provide detailed $p$-adic analysis of the periodic function $g_{k,a,b}$. Finally, we determine the smallest period of $g_{k,a,b}$. Our result extends the Farhi-Kane theorem from the set of positive integers to general arithmetic progressions.

math.NT↗
arXiv2009

Squares in (2^2-1)...(n^2-1) and p-adic valuation

In this paper, we determine all the squares in the sequence $\{\prod_{k=2}^n(k^2-1)\}_{n=2}^\infty $. From this, one deduces that there are infinitely many squares in this sequence. We also give a formula for the $p$-adic valuation of the terms in this sequence.

math.NT↗
arXiv2009

Nair's and Farhi's identities involving the least common multiple of binomial coefficients are equivalent

In 1982, Nair proved the identity: ${\rm lcm}({k \choose 1}, 2{k \choose 2}, >..., k{k \choose k}) ={\rm lcm}(1, 2, ..., k), \forall k\in \mathbb{N}.$ Recently, Farhi proved a new identity: ${\rm lcm}({k \choose 0}, {k \choose 1}, >..., {k \choose k}) =\frac{{\rm lcm}(1, 2, ..., k+1)}{k+1}, \forall k\in \mathbb{N}.$ In this note, we show that Nair's and Farhi's identities are equivalent.

math.NT↗
arXiv2009

Further Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,u_n} and let a >= 2 be any integer. In this paper, we show that, for integers alpha,r >= a and n >= 2*alpha*r, we have L_n >= u_0*r^{alpha+a-2}*(r+1)^n. In particular, letting a = 2 yields an improvement to the best previous lower bound on L_n (obtained by Hong and Yang) for all but three choices of alpha,r >= 2.

math.NT↗
arXiv2008

Infinite divisibility of Smith matrices

Given an arithmetical function $f$, by $f(a, b)$ and $f[a, b]$ we denote the function $f$ evaluated at the greatest common divisor $(a, b)$ of positive integers $a$ and $b$ and evaluated at the least common multiple $[a, b]$ respectively. A positive semi-definite matrix $A=(a_{ij})$ with $a_{ij}\ge 0$ for all $i$ and $j$ is called infinitely divisible if the fractional Hadamard power $A^{\circ r}=(a_{ij}^r)$ is positive semi-definite for every nonnegative real number $r$. Let $S=\{x_1, ..., x_n\}$ be a set of $n$ distinct positive integers. In this paper, we show that if $f$ is a multiplicative function such that $(f*μ)(d)\ge 0$ whenever $d|x$ for any $x\in S$, then the $n\times n$ matrices $(f(x_i, x_j))$, $(\frac{1}{f[x_i, x_j]})$ and $(\frac{f(x_i, x_j)}{f[x_i, x_j]})$ are infinitely divisible. Finally we extend these results to the Dirichlet convolution case which produces infinitely many examples of infinitely divisible matrices. Our results extend the results obtained previously by Bourque, Ligh, Bhatia, Hong, Lee, Lindqvist and Seip.

math.NT↗