All Questions
Tagged with combinatorics number-theory
1,985
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A question on generalized bases
I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
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Let $f$ satisfy $f(mr)<f(r)$ for all $m,r\in \mathbb N$. Is $f(k)$ decreasing for all large $k$?
Consider the question
Let $f:\mathbb N \to \mathbb R$ satisfy $f(mr)<f(r)$ for all $m,r\in \mathbb N$, $m>1$. Is $f(k)$ decreasing for all $k>k_0$ for some $k_0$?
The answer is clearly no ...
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Closed Formula Expression for Sum of Combinatorics
I have recently been interested in the problem of summing Combinatorials. I have been beating my brain for the past days to figure out how to find an explicit closed form of:
$n \choose 0 $+$ n \...
- 54.3k
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Exploring the Intersection of Expander Graphs, Number Theory, Representation Theory and Recent Computer Science Developments
I have a solid understanding of the basics of expander graphs and their properties and the recent development of High-Dimensional Expanders and their application to Random Walks, along with other ...
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When the product of dice rolls yields a square
Succinct Question:
Suppose you roll a fair six-sided die $n$ times.
What is the probability that the product of the rolls is a square?
Context:
I used this as one question in a course for elementary ...
- 239k
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how many natural numbers require at least 6 terms to express as the sum of distinct squares?
I wrote a computer program as an exercise in dynamic programming. It finds the minimum number of distinct squares which sum to some positive target integer n. I found something interesting and would ...
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Existence of a Subset $S$ of $\mathbb{N}$ Where All but Finitely Many Natural Numbers Are Sums of Consecutive Elements of $S$
I am pondering a question in number theory that touches upon the representation of natural numbers as sums of consecutive elements from a subset S of $\mathbb{N}$. Specifically, the question is:
Does ...
- 93.3k
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How many $k$-full numbers are there?
Recall that $n\in\mathbb{N}$ is $k$-full if, for a prime number $p$ : $p\mid n$ implies $p^{k}\mid n$. In the paper I am currently reading it is said that there are $O(B^{1/k})$ $k$-full positive ...
- 78.4k
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On thickness of binary polynomials
OEIS A169945 introduces the concept of thickness of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(...
- 9,592
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Number of Salem–Spencer subsets of $\{1,2,3,\dots ,n\}$
I was wondering about sets that do not contain any $3$-term AP, and came to know that the official name of such a set is Salem–Spencer set. I was considering the question of counting the number of ...
- 9,080
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Quadratic Nimber Equation
I'd like to ask how to solve the quadratic nimber equation $x\otimes x \oplus b \otimes x \oplus c=0$, where $\otimes$ is nim multiplication and $\oplus$ is nim addition.
- 78.4k
5
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Conjecture: $\binom{n}{k } \mod m =0$ for all $k=1,2,3,\dots,n-1$ only when $m $ is a prime number and $n$ is a power of $m$
While playing with Pascal's triangle, I observed that $\binom{4}{k } \mod 2 =0$ for $k=1,2,3$,and $\binom{8}{k } \mod 2 =0$ for $k=1,2,3,4,5,6,7$ This made me curious about the values of $n>1$ and ...
- 6,565
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Non negative integral solutions when coefficients are involved and inequality among parameters
What is the number of non negative integral solutions of
$5w+3x+y+z=100$, such that $w≤x≤y≤z$.
I have the answer key to this but I do not understand how to even start. The second inequality condition ...
- 42.8k
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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 ...
- 6,565
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Sum-free sequence but multiset
The question is:
Show that if $S$ is a set of natural numbers such that no number in S can be expressed as a sum of other (not necessarily distinct) numbers in S, then $\sum_{ s \in S} \frac{1}{s} \...
- 70.4k