Divisibility of the central binomial coefficient $\binom{2n}{n}$

We use the notion of subprime factorization to establish recurrence relations for the number of binomial coefficients in a given row of Pascal’s triangle that are divisible by p^j and not divisible by p^{j+1}, where p is a prime. Using these relations to compute this number can provide significant savings in the number of computational steps.

arXiv: Number Theory, 2019

In this paper, we first prove that for any strong divisibility sequences $\boldsymbol{a} = \left(a_n\right)_{n\geq 1}$, we have the identity: $\mathrm{lcm} \left\lbrace \binom{n}{0}_{\bf{a}}, \binom{n}{1}_{\bf{a}},\dots, \binom{n}{n}_{\bf{a}} \right\rbrace = \frac{\mathrm{lcm} \left(a_1,\dots , a_n , a_{n+1}\right)}{a_{n+1}}$ $\left(\forall n \geq 1\right)$, generalizing the identity of Farhi (obtained in 2009 for $a_n=n$). Then, we derive from this one some other interesting identities. Finally, we apply those identities to estimate the least common multiple of the consecutive terms of some Lucas sequences. Denoting by $\left(F_n\right)_n$ the usual Fibonacci sequence, we prove for example that for all $n \geq 1$, we have \[ \Phi^{\frac{n^2}{4}-\frac{9}{4}} \leq \mathrm{lcm}\left(F_1,\dots,F_n\right) \leq \Phi^{\frac{n^2}{3}+\frac{4n}{3}} , \] where $\Phi$ denotes the golden ratio.

Research in Number Theory, 2021

Let {U n } n≥0 be a Lucas sequence. We show that if U n is a product of factorials, then n ∈ {1, 2, 3, 4, 6, 8, 12}. Further we show that if U n is a product of Catalan numbers or the middle binomial coefficients, then n ∈ {1, 2, 3, 4, 6, 8, 12, 16}. Here the m−th middle binomial coefficient is B m = 2m m and the m−th Catalan number is C m = 1 m+1 2m m. Alternatively, they can be defined recursively as U 0 = 0, U 1 = 1, V 0 = 2, V 1 = r and both recurrences U n+2 = rU n+1 + sU n and V n+2 = rV n+1 + sV n hold for all n ≥ 0. Let PF := {±1} ∪ ± k j=1 m j ! : 1 < m 1 ≤ m 2 ≤ • • • ≤ m k and k ≥ 1 be the set of integers whose absolute values are the product of factorials > 1. Members of PF are sometimes called Jordan-Polya numbers and several recent papers investigate their arithmetic properties. It was shown in [2] that U n ∈ PF implies n < 62000. We reduce this bound to n ∈ {1, 2, 3, 4, 6, 8, 12} in the following theorem.

2013

Given an integer n ≥ 2, let p m (n) denote the middle prime factor of n. We obtain an estimate for the sum of the reciprocals of p m (n) for n ≤ x.

Transactions of the American Mathematical Society, 1986

The number Ψ ( x , y ) \Psi (x,y) of integers ≤ x \leq x and free of prime factors &gt; y &gt; y has been given satisfactory estimates in the regions y ≤ ( log ⁡ x ) 3 / 4 − ε y \leq {(\log x)^{3/4 - \varepsilon }} and y &gt; exp ⁡ { ( log ⁡ log ⁡ x ) 5 / 3 + ε } y &gt; \exp \{ {(\log \log x)^{5/3 + \varepsilon }}\} . In the intermediate range, only very crude estimates have been obtained so far. We close this &quot;gap&quot; and give an expression which approximates Ψ ( x , y ) \Psi (x,y) uniformly for x ≥ y ≥ 2 x \geq y \geq 2 within a factor 1 + O ( ( log ⁡ y ) / ( log ⁡ x ) + ( log ⁡ y ) / y ) 1 + O((\log y)/(\log x) + (\log y)/y) . As an application, we derive a simple formula for Ψ ( c x , y ) / Ψ ( x , y ) \Psi (cx,y)/\Psi (x,y) , where 1 ≤ c ≤ y 1 \leq c \leq y . We also prove a short interval estimate for Ψ ( x , y ) \Psi (x,y) .

We present a general formula for the number of binomial coefficients in a given row of Pascal’s triangle that are divisible by p^j and not divisible by p^{j+1}, where p is a prime.

Journal of Number Theory, 2018

Inventiones Mathematicae, 1984

The American Mathematical Monthly, 2018

We define a new class of integer partitions generalizing hyperbinary partitions. We then utilize these partitions to provide a new, relatively simple formula for the number of binomial coefficients in a particular row of Pascal's triangle that are divisible by a fixed power of a prime. Our formula specializes to a famous result due to N. J. Fine from 1947. Furthermore, we demonstrate a connection between the binomial coefficients congruent to zero modulo a fixed prime power and the so-called digital dominance orders on the natural numbers. A classic problem in combinatorial number theory asks, for a fixed prime number p and a given integer n, how many binomial coefficients of the form n k are divisible by p. We visualize this problem by shading Pascal's triangle so that an entry n k is unshaded (white) if it is not divisible by p and it is shaded (blue) if it is divisible by p. For instance, the upper left image in Figure shows the picture for p = 2. Kung [10] appears to be the first author to include this type of image as a method of visualizing Pascal's triangle modulo a prime. This question naturally extends to divisibility by a fixed prime power, p m , in Pascal's triangle. The remaining three images in Figure show Pascal's triangle reduced modulo 4, 8, and 16, again where we shade the entries divisible by the appropriate power of 2. Of course, the corresponding problem arising from these diagrams is to find a formula for the number of shaded entries in a given row. As luck would have it, the literature abounds with details about and formulas for this quantity. Fine [7] utilizes Lucas's theorem to enumerate the number of entries in row n of Pascal's triangle that are nonzero modulo p. Carlitz [3] gives recursive formulas and generating functions for these counts when p m is fixed, and Everett [5] gives a closed form for the functions introduced by Carlitz in terms of binary words. Furthermore, Rowland and Everett [6] give alternative interesting and efficient formulas for computing the number of entries in row n of Pascal's triangle congruent to 0 mod p m . Finally, Spiegelhofer and Wallner have recently answered a variety of questions posed by Rowland [13]; the results in those papers can be seen as generalizations of Fine's result. In the current note, we add another chapter to this saga by providing a formula for the number of entries congruent to 0 mod p m in row n of Pascal's triangle that uses a certain family of integer partitions. Our connection between Pascal's triangle, these partitions, and the formula for the number of zero entries appears to have been missed in the vast literature about Pascal's triangle mod p m and requires only a simple, relatively nontechnical proof. Moreover, our results can be interpreted in terms of a family of partial orders defined on the natural numbers, which are known as base-p digital dominance orders (see ).

J. Integer Seq., 2014

We give a systematic view of the asymptotic expansion of two well-known sequences, the central binomial coefficients and the Catalan numbers. The main point is explanation of the nature of the best shift in variable n, in order to obtain “nice” asymptotic expansions. We also give a complete asymptotic expansion of partial sums of these sequences.