# 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 ...
• 161
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 ...
• 1,375
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 ...
• 87
1 vote
41 views

### 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 $n$? ...
• 1,811
1 vote
85 views

### 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,...
• 1,250
117 views

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

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 $A+A\subseteq A\times A$ ?
• 1,170
199 views

### 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 ...
• 26.6k
255 views

### 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 ...
• 165
1 vote
34 views

### 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 ...
• 401
72 views

### 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) ...
• 331
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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 ...
406 views

### Probability that the sum of $k$ distinct integers selected from $1, 2, \dots, n$ is divisible by $n$

Would you please help me solve Problem 7 of Section 4.5, "Combinatorial Number Theory," in An Introduction to the Theory of Numbers, Niven, Zuckerman, Montgomery, 5th ed., Wiley (New York), 1991: ...
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### 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 ...