arXiv subjects

Phakhinkon Phunphayap

Publications and source records attributed to Phakhinkon Phunphayap.

2 recordsLinked to original sources

arXiv2019

Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients II

In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients ${p^a n \choose n}_F$ for all primes $p$ and positive integers $a$ and $n$. This is a continuation from our previous article extending some results in the literature, which deal only with $p = 2,3,5,7$ and $a = 1$. Then we use these formulas to characterize the positive integers $n$ such that ${pn \choose n}_F$ is divisible by $p$, where $p$ is any prime which is congruent to $\pm 2 \pmod{5}$.

math.NT↗

arXiv2018

Reciprocal Sum of Palindromes

A positive integer $n$ is said to be a palindrome in base $b$ (or $b$-adic palindrome) if the representation of $n = (a_k a_{k-1} \cdots a_0)_b$ in base $b$ with $a_k \neq 0$ has the symmetric property $a_{k-i} = a_i$ for every $i=0,1,2,\ldots ,k$. Let $s_b$ be the reciprocal sum of all $b$-adic palindromes. It is not difficult to show that $s_b$ converges. In this article, we obtain upper and lower bounds for $s_b$ and the inequality $s_{b} <s_{b'}$ for $2\leq b<b'$. Its consequences and some numerical data are also given.

math.CA↗