
Prapanpong PongsriiamSilpakorn University · Department of Mathematics
Prapanpong Pongsriiam
Ph.D. Mathematics
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93
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Introduction
Prapanpong Pongsriiam currently works at the Department of Mathematics, Silpakorn University, Thailand. His main research interest is in number theory (analytic, combinatorial, elementary). Topics of particular interest are
1. Problems in the divisor function,
2. Fibonacci numbers and the order of appearance,
3. Factorials and binomial coefficients,
4. The floor function,
5. Digital problems.
He sometimes does research on metric-preserving functions. His Erdos number is 2.
Skills and Expertise
Additional affiliations
May 2012 - present
Education
August 2006 - May 2012
The Pennsylvania State University
Field of study
- Mathematics
June 2004 - August 2006
Publications
Publications (93)
Let $d(n)$ be the number of positive divisors of $n$, and let $c_r(a)$ be the Ramanujan's sum. We prove that for $q\geq 1$, $a\in \mathbb Z$, and $x\geq 1$,
$$
\sum_{\substack{n\leq x\\ n\equiv a\bmod q}}d(n) = \frac{x}{q} \sum_{r|q} \frac{c_r(a)}{r} \left(
\log\frac{x}{r^2} +2\gamma -1
\right) +O\left(
(x^{\frac13}+q^{\frac12})x^{\varepsilon}
\rig...
In this article, we completely determine the length of longest arithmetic progressions in the least positive reduced residue system and in all reduced residue systems modulo $n$ for every positive integer $n$.
Let \(\alpha = (1+\sqrt{5})/2\) be the golden ratio, and let \(B(\alpha ) = (\left\lfloor n\alpha \right\rfloor )_{n\ge 1}\) and \(B(\alpha ^2) = \left( \left\lfloor n\alpha ^2\right\rfloor \right) _{n\ge 1}\) be the lower and upper Wythoff sequences, respectively. In this article, we obtain a new estimate concerning the fractional part \(\{n\alpha...
Lucas sequence of the first kind is an integer sequence $(U_n)_{n\geq0}$ which depends on parameters $a,b\in\mathbb{Z}$ and is defined by the recurrence relation $U_0=0$, $U_1=1$, and $U_n=aU_{n-1}+bU_{n-2}$ for $n\geq2$. In this article, we obtain exact divisibility results concerning $U_n^k$ for all positive integers $n$ and $k$. This extends man...
For each \(s\in {\mathbb {R}}\) and \(n\in {\mathbb {N}}\), let \(\sigma _s(n) = \sum _{d\mid n}d^s\). In this article, we study the number of sign changes in the difference \(\sigma _s(an+b)-\sigma _s(cn+d)\) where a, b, c, d, s are fixed, the vectors (a, b) and (c, d) are linearly independent over \({\mathbb {Q}}\), and n runs over all positive i...
These are the promotional pages of my book:
P. Pongsriiam, Analytic Number Theory for Beginners, Second Edition, American Mathematical Society, 2023
https://bookstore.ams.org/view?ProductCode=STML/103
We introduce a new arc in directed graphs of integers. Among other things, we determine the positive integers that have arcs to all except a finite number of positive integers. We also propose some possible research problems at the end of this article.
The happy function S of each positive integer x is defined to be the sum of the squares of the decimal digits of x . For example, S (2) = 4 and S (123) = 1 ² + 2 ² + 3 ² = 14. It is well known that for any , there exists such that , where S ( n ) is the n -fold composition of S . In addition, if and for some , then x is called a happy number.
Let $ b $ and $ b_1 $ be distinct positive integers larger than $ 1 $, and let $ A_{b} (n) $ and $ A_{b_1} (n) $ be the number of palindromes in bases $ b $ and $ b_1 $ that are less than or equal to $ n $, respectively. In this article, we finish the comparative study of the functions $ A_b (n) $ and $ A_{b_1} (n) $. As a result, we present the fu...
For each positive irrational number α, the Beatty sequence generated by α is given byB(α)=(⌊bα⌋)b≥1. Previously, Pongsriiam and his coauthors determined the structure of the sumsets B(α)+B(α), B(α2)+B(α2)+B(α2), B(α)+B(α2)+B(α2), B(α)+B(α2), and B(α2)+B(α2), where α=(1+5)/2 is the golden ratio. In this article, we extend the investigation to the su...
Let $ \alpha $ be the golden ratio, $ m\in \mathbb N $, and $ B(\alpha^m) $ the Beatty sequence (or Beatty set) generated by $ \alpha^m $. In this article, we give some combinatorial structures of $ B(\alpha^m) $ and use them in the study of associated sumsets. In particular, we obtain, for each $ m\in \mathbb N $, a positive integer $ h = h(m) $ s...
Let $ b \geq 2 $ and $ n \geq 1 $ be integers. Then $ n $ is said to be a palindrome in base $ b $ (or $ b $-adic palindrome) if the representation of $ n $ in base $ b $ reads the same backward as forward. Let $ A_b (n) $ be the number of $ b $-adic palindromes less than or equal to $ n $. In this article, we obtain extremal orders of $ A_b (n) $....
We give a characterization for the integers $ n \geq 1 $ such that the Fibonomial coefficient $ {pn \choose n}_F $ is divisible by $ p $ for any prime $ p \neq 2, 5 $. Then we use it to calculate asymptotic formulas for the number of positive integers $ n \leq x $ such that $ p \mid {pn \choose n}_F $. This completes the study on this problem for a...
Let a and b be positive integers and f an arithmetic function. In this article, we investigate whether or not a certain condition on the value of f implies a = b. For example, if f is the sum of divisors function and f (an) = f (bn) for all positive integers n, then a = b.
Let \(n, q, a\in \mathbb N\), d(n) the number of positive divisors of n, and A(x, q, a) the sum of d(n) over \(n\le x\) and \(n\equiv a\pmod q\). Previously we obtained an asymptotic formula for the second moment $$\begin{aligned} V(x,Q) = \sum _{q\le Q}\sum _{a=1}^q \left| A(x,q,a)-M(x,q,a)\right| ^2, \end{aligned}$$where \(Q\le x\) and M(x, q, a)...
It is well known that arithmetic progressions of squares have length at most 3. In this note, we extend the result to the case of quadratic polynomials.
For each $s\in \mathbb R$ and $n\in \mathbb N$, let $\sigma_s(n) = \sum_{d\mid n}d^s$. In this article, we give a comparison between $\sigma_s(an+b)$ and $\sigma_s(cn+d)$ where $a$, $b$, $c$, $d$, $s$ are fixed, the vectors $(a,b)$ and $(c,d)$ are linearly independent over $\mathbb Q$, and $n$ runs over all positive integers. For example, if $|s|\l...
We study a variation of the Kaprekar operator and the reverse and add operator concerning the number 1089.
Lucas sequences of the first and second kinds are, respectively, the integer sequences $ (U_n)_{n\geq0} $ and $ (V_n)_{n\geq0} $ depending on parameters $ a, b\in\mathbb{Z} $ and defined by the recurrence relations $ U_0 = 0 $, $ U_1 = 1 $, and $ U_n = aU_{n-1}+bU_{n-2} $ for $ n\geq2 $, $ V_0 = 2 $, $ V_1 = a $, and $ V_n = aV_{n-1}+bV_{n-2} $ for...
Let n and k be positive integers and σ(n) the sum of all positive divisors of n. We call n an exactly k-deficient-perfect number with deficient divisors d1, d2,. .. , d k if d1, d2,. .. , d k are distinct proper divisors of n and σ(n) = 2n − (d1 + d2 + · · · + d k). In this article, we show that the only odd exactly 3-deficient-perfect number with...
Chase introduced the concept of digit maps generalizing that of happy functions. We extend the investigation further by considering the compositions of various digit maps. We prove that if F is such a composition and x is any positive integer, then the sequence (F (n) (x)) n≥0 either converges or eventually becomes a cycle. Furthermore, we show tha...
We explicitly solve the Diophantine equations of the form A 1 A 2 A 3 · · · An ± 1 = B ℓ m where (An) n≥1 and (Bm) m≥1 are the Fibonacci or Lucas sequences and m, n, ℓ are positive integers. This extends some results concerning a Fibonacci version of the Brocard-Ramanujan equation.
Let α be the golden ratio and βα = −1. In the study of sumsets associated with Wythoff sequences, it is important to prove the inequality 0 < {bα} + β n < 1 for integers b and n in a certain range. In this article, we continue the investigation by replacing {bα} + β n by √ 5β n−1 − {bα}.
This contains parts of my book on analytic number theory for beginners, first edition.
A positive integer n is called a palindrome if the decimal representation of n reads the same backward as forward. Domotorp asked and Tao answered that the sequence of palindromes does not contain arbitrarily long arithmetic progressions. From Tao's comments, it is expected that the length of such the progressions should be less than 10⁸. However,...
This is the beamer of my presentation in the online conference RDNT 2020. It is a combination of old and new files.
Let α be the golden ratio and βα = −1. In the study of sumsets associated with Wythoff sequences, it is important to prove the inequality 0 < {bα} + β n < 1 for integers b and n in a certain range. In this article, we continue the investigation by replacing {bα} + β n by √ 5β n−1 − {bα}.
Let m and n be positive integers and let F m and F n be the mth and nth Fibonacci numbers. In this article, we find the Zeckendorf representation of the multiplicative inverse of F m modulo F n when m is small or when m is closed to n and (m, n) ≤ 2.
In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients $\binom{p^a n}{n}_F$ 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 char...
This is a draft of my book containing the cover of the book, the whole Chapters 1 and 2, two pages of Chapter 3 to 10, references, and indices of the book.
We introduce new classes of functions related to metric-preserving functions, b-metrics, and ultrametrics. We investigate their properties and compare them to those of metric-preserving functions.
Let $n$ and $k$ be positive integers and $\sigma(n)$ the sum of all positive divisors of $n$. We call $n$ an exactly $k$-deficient-perfect number with deficient divisors $d_1, d_2, \ldots, d_k$ if $d_1, d_2, \ldots, d_k$ are distinct proper divisors of $n$ and $\sigma (n)=2n-(d_1+d_2+\ldots + d_k)$. In this article, we show that the only odd exactl...
A positive integer n is a b-adic palindrome if the representation of n in base b reads the same backward as forward. Let s b be the reciprocal sum of all b-adic palindromes. In this article, we obtain upper and lower bounds, and an asymptotic formula for s b. We also show that the sequence (s b) b≥2 is strictly increasing and log-concave.
Upper and lower bounds, and asymptotic formulas for the reciprocal sum of all b-adic palindromes. Some properties of the sequence of the reciprocal sum of b-adic palindromes.
Explain some results concerning arbitrarily long arithmetic progressions in integer sequences.
Let Fn and Ln be the nth Fibonacci and Lucas numbers, respectively. Let ω(n) be the number of prime factors of n, d(n) the number of positive divisors of n, A(n) the least positive reduced residue system modulo n, and (n) the length of the longest arithmetic progressions contained in A(n). On the occasion of attending the 18th Fibonacci Conference,...
This contains the estimates for the reciprocal sums of b-adic palindromes for b up to 100. This data is to supplement our article on the reciprocal sum of palindromes which will appear in the Journal of Integer Sequences sometime in October-December 2019.
Let Fn and Ln be the nth Fibonacci and Lucas numbers, respectively. Let φ(n) be the Euler totient function of n and σ k (n) the sum of kth power of the positive divisors of n. Luca obtains the inequalities φ(Fn) ≥ F φ(n) , σ0(Fn) ≥ F σ 0 (n) , and σ k (Fn) ≤ F σ k (n) for all n, k ≥ 1. In this article, we extend Luca's result by replacing the funct...
In this article, we give explicit formulas for the p-adic valuations of the Fibonomial coefficients (p a n 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 posit...
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 cha...
Hermite's identity states that ∑ 0≤k≤n−1 ⌊ x + k n ⌋ = ⌊nx⌋ for x ∈ R and n ∈ N. In this article, we give a generalization of this identity and show some applications. For example, we consider the above sum when k ranges over the integers from a to b, where a < b are integers. Then we apply it to give another proof of a recent result of Tverberg. W...
Analogue of primes in arithmetic progressions and arithmetic progressions of primes for other integer sequences.
Previously, we investigated some relations between b-metrics and metric-preserving functions. In this article, we continue the investigation by giving a solution to a problem we left open in the previous article. In addition, there are some results in the literature which involve the concept of b-metric and inframetric (or weak-ultrametric). We sho...
Let n ≥ 0 and b ≥ 2 be integers. Then n is said to be a palindrome in base b (or b-adic palindrome) if n = 0 or n ≥ 1 and the representation of n = (a k a k−1 · · · a1a0) b in base b with a k ̸ = 0 has the symmetric property a k−i = ai for 0 ≤ i ≤ k. Let A b (m) be the number of b-adic palindromes not exceeding m. In addition, let A (even) b (m) an...
A brief history and some old and new results on primes and Dirichlet's divisor problem over residue classes.
Recent and complete results on the exact divisibility by powers of the Fibonacci and Lucas numbers
The widest explicit formulas for the p-adic valuations of Fibonomial coefficients.
Let Fn be the nth Fibonacci number. For each positive integer m, the order of appearance of m, denoted by z(m), is the smallest positive integer k such that m divides Fk. Recently, D. Marques has obtained a formula for z(FnFn+1), z(FnFn+1Fn+2), and z(FnFn+1Fn+2 Fn+3). In this paper, we extend Marques’ result to the case z(FnFn+1 ··· Fn+k), for 4 ≤...
In 2014, Pongsriiam obtained the results on exact divisibility by powers of the Fibonacci and Lucas numbers. For instance, he proved that if Fkn || m, n ≥ 3, and n ≢ 3 (mod 6), then Fk+1n || Fnm. In this article, we give the converse of those theorems.
In this work, we propose new methods based on the root-finding methods in numerical analysis to calculate the inverse of an integer a modulo N for any positive integer N. We apply Newton’s and secant methods with a power-reduction process and Newton’s and secant methods together with the binary representation and Zeckendorf representation to determ...
We introduce new classes of functions related to metric-preserving functions and b-metrics. We investigate their properties and compare them to those of metric-preserving functions.
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 such that \(m\mid F_k\). In this article, we obtain a complete formula for the p-adic valuations of z(m!) for all \(m\in {\mathbb {N}}\) and apply it to solve some Diophantine equations i...
This is from the talk I gave at MUIC on September 19, 2018.
Let Fn and Ln be the nth Fibonacci number and Lucas number, respectively. The order of appearance of m in the Fibonacci sequence, denoted by z(m), is the smallest positive integer k such that m divides F k. The formula for z(LnL n+1 L n+2 · · · L n+k) has been recently obtained by Marques for 1 ≤ k ≤ 3, and by Marques and Trojovsk´yTrojovsk´y for k...
On Fibonacci and Lucas numbers which have exactly three prime factors and unique properties of F_18 and L_18.
This is the pdf file of my presentation in the 18th Fibonacci Concference at Dalhousie University, Halifax, Nova Scotia, Canada during July 1-7, 2018.
This file contains the data which I used in the classification of the Fibonacci and Lucas numbers which have exactly three prime factors. The paper ( in preparation) will be presented in the 18th Fibonacci conference at Dalhousie University, Halifax, Nova Scotia, Canada, sometime on July 1-7, 2018.
We obtain explicit formulas for the p-adic valuations of Fibonomial coefficients which extend many results in the literature.
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 $s_b$ c...
Let $F_{n}$ and $L_n$ be the $n$th Fibonacci and Lucas number, respectively. For each positive integer $m$, the order of appearance of $m$ in the Fibonacci sequence, denoted by $z(m)$, is the smallest positive integer $k$ such that $m$ divides $F_k$. Recently, D. Marques has obtained a formula for $z(F_{n}F_{n+1})$, $z(F_{n}F_{n+1}F_{n+2})$, and $z...
We solve the diophantine equations of the form $A_{n_1}A_{n_2}\cdots
A_{n_k}\pm 1 = B_m^2$ where $(A_n)_{n\geq 0}$ and $(B_m)_{m\geq 0}$ are either
the Fibonacci sequence or Lucas sequence. This extends the recent result of
Marques (2011) and Szalay (2012) published in Portugaliae Mathematica.
Hong recently explored when the value of the generating function of the Fibonacci sequence is an integer. He noticed a pattern and raised some questions about it. Here, we completely answer those questions.
We explicitly solve the Diophantinc equations of the form (Equation presented) where (An) and (Bm) are the Fibonacci or Lucas sequences.
We study the sums introduced by Jacobsthal and Tverberg and show that the extreme values of the sums are connected with Jacobsthal and Jacobsthal-Lucas numbers.
Let Fn and Ln be the nth Fibonacci number and Lucas number, respectively. The order of appearance of m in the Fibonacci sequence, denoted by z(m), is the smallest positive integer k such that m divides Fk. Marques obtained the formula of z(Lnk) in some cases. In this article, we obtain the formula of z(Lnk) for all n, k≥1.
We show that a Fibonacci number Fm can be written as a product of Lucas numbers if and only if m = 2^k or m = 3 ⋅ 2^k for some k ≥ 0. Similar results are also given.
Let f(n) be the number of relatively prime subsets of {1, 2, 3,…,n}. In this article, we obtain some local behaviors of f(n) and related functions. For instance, we show that (f(n))2 − f(n − 1)f(n + 1) is positive for every odd number n ≥ 3, and f(6n+2) f(6n+1) > f(6n+4) f(6n+3) > f(6n+6) f(6n+5) < f(6n+8) f(6n+7) for every n ≥ 2.
We give new results on exact divisibility by powers of the Fibonacci and Lucas numbers. For example, we prove that if $F_n^{k}$ exactly divides $m$ and $n$ is not congruent to $3$ modulo $6$, then $F_n^{k+1}$ exactly divides $F_{nm}$. We also provide some examples and open questions.
Kirk and Shahzad have recently given fixed point theorems concerning local
radial contractions and metric transforms. In this article, we replace the
metric transforms by metric-preserving functions. This in turn gives several
extensions of the main results given by Kirk and Shahzad. Several examples are
given. The fixed point sets of metric transf...
We extend the definition of weak symmetric continuity to be applicable for
functions defined on any nonempty subset of $\R$. Then we investigate basic
properties of weakly symmetrically continuous functions and compare them with
those of symmetrically continuous functions and weakly continuous functions.
Several examples are also given.
Let $F_n$ be the $n$th Fibonacci number. Let $m, n$ be positive integers.
Define a sequence $(G(k,n,m))_{k\geq 1}$ by $G(1,n,m) = F^m_n$, and $G(k+1,n,m)
= F_{nG(k,n,m)}$ for all $k\geq 1$. We show that $F_n^{k+m-1}\mid G(k,n,m)$ for
all $k, m, n\in\mathbb N$. Then we calculate
$\frac{G(k,n,m)}{F_n^{k+m-1}}\pmod{F_n}$.
Four functions counting the number of subsets of {1, 2, …, n} having particular properties are defined by Nathanson and generalized by many authors. They derive explicit formulas for all four functions. In this paper, we point out that we need to compute only one of them as the others will follow as a consequence. Moreover, our method is simpler an...
Functions whose composition with every metric is a metric are said to be
metric-preserving. In this article, we investigate a variation of the concept
of metric-preserving functions where metrics are replaced by ultrametrics.
Let F n be the nth Fibonacci number. The period modulo m, denoted by s(m), is the smallest positive integer k for which F n+k ≡F n (modm) for all n≥0. In this paper, we find the period modulo product of consecutive Fibonacci numbers. For instance, we prove that, for n≥1, s(F n F n+1 F n+2 F n+3 )=n(n+1)(n+2)(n+3),ifn¬≡0(mod3),2n((n+1)(n+2)(n++3) 3,...
For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$
denote the greatest common divisor of the elements of $A$. Let
$f\left(n\right)$ and $\Phi\left(n\right)$ denote, respectively, the number of
subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\right) =
1$ and the number of subsets $A$ of $\left\{1, 2, \...
Four functions counting the number of subsets of $\{1, 2, ..., n\}$ having
particular properties are defined by Nathanson and generalized by many authors.
They derive explicit formulas for all four functions. In this paper, we point
out that we need to compute only one of them as the others will follow as a
consequence. Moreover, our method is simp...
We investigate the properties of uniformly symmetrically continuous functions
and compare them with those of symmetrically continuous functions and uniformly
continuous functions. We obtain some characterizations of uniformly
symmetrically continuous functions. Explicit examples are also given.
appears also in Ramanujan's Lost Notebook Part IV by Professor Andrews and Professor Berndt
available in amazon at http://www.amazon.com/distribution-divisor-function-arithmetic-progressions/dp/3659239445/ref=sr_1_1?ie=UTF8&qid=1389350294&sr=8-1&keywords=The+distribution+of+the+divisor+function
Thesis (M.Sc.)--Chulalongkorn University, 2006 It is an interesting problem to study the natural action by automorphisms on the unitary dual of a group G. In this thesis, we compute the kernel of the action and determine whether it is equal to Inn G or Aut G in a number of cases. การศึกษาการกระทำที่เป็นธรรมชาติโดยอัตสัณฐานบนคู่ยูนิแทรีของกรุปเป็นปั...
For a 2 A µ R; let La(A) and Ua(A) denote respectively the set of all strictly increasing sequences in A converging to a and the set of all strictly decreasing sequences in A converging to a: The function f : A ! R is said to be weakly continuous at a 2 A if (i) La(A) 6= ? implies (f(xn)) converges to f(a) for some (xn) in La(A) and (ii) Ua(A) 6=?...
A real-valued function f on a nonempty subset A of ℝ is said to be symmetrically continuous at a∈A if for every ε>0 there is a δ>0 such that |f(a+h)-f(a-h)|<ε whenever h∈ℝ, |h|<δ a+h∈A and a-h∈A. It has been shown by the authors that products, quotients and compositions of functions do not preserve symmetric continuity. In this paper, some remarkab...