# Questions tagged [combinatorial-number-theory]

This is a tag used for number-theoretical questions with combinatorial flavor.

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### Examples in number theory where a heuristic argument fails

Many conjectures in number theory are motivated by heuristic arguments, and many results that are known to be true can be predicted by heuristic arguments. To give an example, consider the Euler ...
1 vote
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### Does every positive power of $7$ contain $2$ in their ternary representation?

Does every positive power of $7$ contain $2$ in their ternary representation? I.e. Determine whether there exists a finite subset of non-negative integers $\{a_1 < \dots < a_n \}$ and a positive ...
133 views

### N*n*m distinguishable balls with m different colors

I recently asked the following question which is resolved: n*m distinguishable balls with m different colors, the probability of randomly choosing k balls containing all balls from at least 2 ...
1 vote
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### Simplify $\frac{1}{n}\sum_{d\mid u} \varphi(d)\cdot 2^{\frac{nr}{d}}$

I have been trying to solve the following problem: Let $n$ and $r$ be positive integers. How many subsets of $\lbrace 1,2,\dots, nr\rbrace$ are there whose elements have a sum divisible by $n$? ...
1 vote
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### Proving a monotonic subsequence exists

CONTEXT I have been studying about finding the limit of a sequence lately, and it becomes apparent that one way for a sequence to have limit is that it is monotonic and has a lower (or upper) bound. I,...
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### Prove prime $p$ divides $\binom{p^k}{n}$ for any $n$, $1 \le n \le p^k-1$, $k$ a positive integer

I have got $$\binom{p^k}{n} = \cfrac{p^k}{n} \binom{p^k-1}{n-1}=p\biggl[\cfrac{p^{k-1}}{n}\binom{p^k-1}{n-1}\biggl]$$ The first approach is to show that the term in the square bracket is also an ... 420 views

Let $m>1$ and $p_2,\cdots,p_k$ are all odd prime numbers less than $2m$. $q,a_2,\cdots,a_k$ are arbitrarily selected integers(I mean no matter how you choose these numbers). $$G_1=\lbrace n\in \... 2 votes 0 answers 156 views ### Conjecture that relates the set of primes to inequalities The following conjecture which I posed has been verified for all n<10000 , my question is to find a proof or a disproof of it. The conjecture is as follows: Consider for all n \in \mathbb N , ... 1 vote 1 answer 83 views ### Ann and Ben plays the following game My Math Team Teacher gave us this combinatorial-number-theory problem. Ann and Ben plays the following game: The numbers 1, 2, 3,...,n, where n\in\mathbb{Z}^+, are written on the blackboard. Ann ... 5 votes 1 answer 81 views ### Who has the winning strategy for the game? My friend Mickey asked me to play the following game with him. The number 1 is written on the whiteboard. Me and Mickey take turns to do the following, starting with me. If the number written on ... 2 votes 3 answers 233 views ### A fraction p/q<1 such that p+q=333. Let p/q<1 and p+q=333. Show that the number of fractions where p and q has no common factor is 108. This is what I’ve worked out: 333/2 = 166.5, and since p<q, p\leq 166,  ... 2 votes 1 answer 92 views ### How many copied numbers? When I was doing my math team training, I encountered a difficult question. The question is If there is exist a natural number n such that a number equal n written twice, then n is a copied ... 1 vote 1 answer 57 views ### Prove that for every n\geq6 the equation 1/a_1^2 + ... 1/a_n^2 = 1 has answer in \mathbb{Z} Prove that for every n \geq 6 the equality 1/a_1^2 + ... 1/a_n^2 = 1 has answer in \mathbb{Z} (I mean it has an answer where all a_i \in \mathbb{Z}) (Repetition is allowed) After some testing ... 5 votes 1 answer 109 views ### Is the problem Find all x,y \in \mathbf{N} such that \binom{x}{2} = \binom{y}{5} solved? I was recently browsing and came upon this document which gives some open problems involving Diophantine Equations. Document: http://www.math.leidenuniv.nl/~evertse/07-workshop-problems.pdf Upon ... 2 votes 2 answers 131 views ### The number of f \circ f=f  Let A=\{1,2,3...n\}，How many functions satisfy the following conditions？ (1) f \circ f=f  (2) f \circ f =I_A (3) f \circ f\circ f=I_A \circ is composite. The question is from a '... 3 votes 0 answers 45 views ### If A\subset\mathbb N does not contain any two integers such that one divides the other, must A have density 0? Suppose we have a set A\subset\{1,2,3,\ldots\} such that there do not exist m,n\in A where m\mid n. Does it follow that A has natural density 0? (Note: The definition of the natural density ... 1 vote 0 answers 96 views ### Generating function for number of partitions with Rank and Crank The generating function for the number of partitions, p(n) of a number is given by \displaystyle{\sum_{n=1}^{\infty}p(n)q^n=\frac{1}{(q;q)_{\infty}}} where (q;q)_{\infty}=\displaystyle{\lim_{n \... 0 votes 0 answers 41 views ### What composite number can be represented as \binom{n}{k} for k\neq1 or n-1 This is a problem I came up with when I was working with binomial coefficients. Let's call the title statement (*). Obviously for any primes do not satisfy (*), therefore we only need to focus on ... 5 votes 2 answers 287 views ### Number of triples of divisors who are relatively prime as a triple Given a number n \in \mathbb{N}, define a(n)=\{(d_1,d_2,d_3): d_i|n,\ \gcd(d_1,d_2,d_3)=1,\ 1 \leq d_1 \leq d_2 \leq d_3\}. What is |a(n)|? There were similar questions asked about pairs rather ... 0 votes 0 answers 98 views ### Is this a proof for the rational distance problem of a unit square? I found this 2015 research paper (http://unsolvedproblems.org/S75.pdf) that claims it has solved the rational distance problem of a unit square. When I look in other stackexchange and reddit threads, ... 1 vote 0 answers 116 views ### Combinatorial counting problem for induced subgraphs We define P(x,y) in the following way: Given an initial undirected graph of n vertices, P(x,y)=1 if an induced subgraph with exactly x vertices and y edges exists. Else P(x,y)=0. For example, let ... 1 vote 1 answer 161 views ### \sum_{i=1}^n P(n,i) [duplicate] We can know clearly from$$(1+X)^n=\sum\limits_{i=0}^{n}C(n,i)X^n$$that$$ \sum\limits_{i=0}^{n}C(n,i)=2^n. Whereas, I want to know if there are any researched results about permutations in the ... 92 views

### $A+A\subseteq A\times A$

Define $A+A=\{a+b\colon a,b\in A\}$,$A\times A =\{ab\colon a,b\in A\}$. Does there exist a finite integer set $A\subseteq \mathbb{Z}^+$, such that $|A|>1$ and $A+A\subseteq A\times A$ ?
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### Bijective proof that $8+1=9$, or really $3^2-1=2^3$

Catalan's conjecture states that $8$ and $9$ are the only consecutive powers. This suggests to me that the identity $3^2-1=2^3$ might be purely "accidental". So here's the challenge: Is there any ...
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### Lonely Runner Conjecture proof for $k=3$ runners

I'm currently reading Lonely runner conjecture which has been proved upto $k=7$. I could understand the cases with $k=1,2$ i.e, one and two runners repsectively, but I am stuck at $k=3$. Can anyone ...
1 vote
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### Finding number of n digit Terminate NUMBERS

One is allowed to use the digits 5,6,7,8 any number of times to form a terminate number. A n digit terminate number is one which contains digit 5 either an even number of times or does not contain ...
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### moving on a grid with integer condition

assume you are a flea jumping on a grid (similar to $\mathbb{Z}^2$). You can do any jump that define an integer distance, but moreover it has to change both coordinates (for instance, the moves (3,4) ...
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### Relation between some prime numbers and their sum

Let $p$ and $q$ be two prime numbers less than $900$ billion. If $p + 6, p + 10, q + 4, q + 10$ and $p + q + 1$ are all primes, what is the greatest value that $p + q$ can take? Can you help me to ...
### Multisets $A$ and $B$ of positive integers so that sum of $A$ is product of $B$ and vice versa
I happened to notice that $2 \cdot 3 = 1+5$ and $2+3 = 1 \cdot 5$. I researched a little further: With size (cardinality) 1 there is trivially an infinite amount of solutions. With size 2 there is ...