All Questions
60
questions
1
vote
0
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322
views
A conjecture on representing $\sum\limits_{k=0} ^m (-1)^ka^{m-k}b^k$ as sum of powers of $(a+b)$.
UPATE: I asked this question on MO here.
I was solving problem 1.2.52 in "An introduction to the theory of numbers by by Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery"
Show that if ...
- 6,620
9
votes
8
answers
910
views
Why is the binomial coefficient $\binom{\text{even}}{\text{odd}}$ always even?
I recently came across the following:
Fix an odd $k\in\mathbb{N}$. Then for all even $n\in\mathbb{N}, \binom{n}k \text{ is even.}$
I'd like to complete my attempted proof by induction (see below), ...
- 183
4
votes
1
answer
170
views
Greatest common divisor of some binomial coefficients
Note that this is now cross-posted on mathoverflow.
While making some computation, I stumbled upon a curious relation among some binomial coefficients.
Consider the sequence of binomial coefficients $...
- 2,103
1
vote
1
answer
72
views
Binomial sum closed formula [closed]
I was wondering if a closed formula, with few terms in $n$ and $k$, is known for
$$f(n,k):=\sum\limits_{i=1}^k\left(\begin{array}{c}nk \\ ni\end{array}\right),$$
for arbitrary positive integers $n$ ...
- 415
4
votes
0
answers
88
views
Series expansion in binomial coefficients
After noticing that the first three terms of OEIS A016269 coincide with those of OEIS A030053 and the fourth differs only by $1$, and working with successive approximations, I obtained the following ...
- 2,103
1
vote
0
answers
62
views
Number of solutions to linear equation $x_1+x_2+\dots+x_n=m$ when the domain of $x_i\ne$ domain of $x_j$
In the lecture notes of one of my previous classes, it was used that if we have an equation of the form
$$\tag{1}
x_1+x_2+\dots+x_n=m
$$
then the total number of solutions, when each $x_i$ is a non-...
- 175
0
votes
1
answer
110
views
Prove that 2 binomial coefficients have a common prime factor
Let a, b, n be integers such that 0 < a ≤ b < n. Prove that there
exists a prime number p that divides both
\begin{equation}\binom{n}{b}, \binom{n}{a}\end{equation}
I have been proving with the ...
5
votes
3
answers
926
views
Prove that ${2p\choose p}\equiv 2\pmod {p^2}.$
Prove that ${2p\choose p}\equiv 2\pmod {p^2}.$
My progress:
Consider any $S={(x,y),~~1\le x\le p,~~1\le y\le 2}.$ Note that $|S|=2p.$
Now consider $A\subseteq S$ such that $|A|=p.$
So there are a ...
- 4,135
0
votes
0
answers
301
views
How to upper-bound the Binomial coefficient with a Polynomial?
I want to upper-bound the following Binomial coefficient with a Polynomial $p(n)\in O(n^c)$:
$$
{n \choose k}=\frac{n!}{k!\cdot\left(n-k\right)!}\leq p(n)
$$
I know that $k\leq n$ but which $p(n)$ I ...
- 1,979
2
votes
1
answer
81
views
Does this summation formula for $\pi(x)$ work out? (Answer: no)
Let $p_i$ be the $i$th prime number. For $x \in \Bbb{R}, x \gt 0$, define:
$$
\Delta(x) =\left \{\prod_{i=1}^r q_i : \{q_1, \dots, q_r \}\subset \{p_1, p_2, \dots, p_{\pi(x)}\}\right\}
$$
Let $\omega$...
2
votes
1
answer
202
views
If $xy = ax + by$, prove the following: $x^ny^n = \sum_{k=1}^{n} {{2n-1-k} \choose {n-1}}(a^nb^{n-k}x^k + a^{n-k}b^ny^k),n>0$
If $xy = ax + by$, prove the following: $$x^ny^n = \sum_{k=1}^{n} {{2n-1-k} \choose {n-1}}(a^nb^{n-k}x^k + a^{n-k}b^ny^k) = S_n$$ for all $n>0$
We'll use induction on $n$ to prove this.
My ...
4
votes
2
answers
79
views
Proving that $(k + 1) \mid \binom{n}{k}\binom{n+1}{k}$ for positive integers $n, k$ [duplicate]
I've been playing around with some binomial coefficients and their divisibility, and I stumbled upon a relation that seems to hold, at least for $0 < k < n < 200$ (checked with Python): $$ (k ...
- 43
8
votes
0
answers
385
views
Making binomial coefficients out of binomial coefficients
For binomial coefficients, we have
$\binom{n}{k}=\frac{n!}{k!(n-k)!}$
Now, fix a positive integer $s$ and define a function $n?=\binom{n}{s} \cdot \binom{n-1}{s} \cdot ... \cdot \binom{s}{s}$ or $1$ ...
- 2,196
2
votes
1
answer
48
views
Prove that $C(x+y,y)$ is compositive number if $x,y\in \mathbb{Z}_{>1}^{+}$
Prove that $$x,y \in \mathbb{Z}^{+}, x,y>1 \implies \binom{x+y}{y} \quad \text{is compositive.}$$
My approach: Let $x,y\in \mathbb{Z}^{+}$ such that $x,y>1$ so we need to prove that $\binom{x+...
user798113
3
votes
1
answer
84
views
Binomial coefficient system of equations
Given that ${n\choose k}=3003$ and ${n\choose k-1}=2002$, find $n$ and $k$.
How can I solve these equations without guessing/WA? I can't fully eliminate one of the variables, so I tried to break down ...
- 191