# elementary-number-theory's questions - Chinese 1answer

23.643 elementary-number-theory questions.

### How do i find all possible ways of obtaining a total value of 40 cents from 5 cent and 8 cent stamps?

You have an inexhaustible supply of 5 cent and 8 cent stamps. List all possible ways of obtaining a total value of 40 cents with these stamps. I used a probability tree to solve this problem. But i ...

### On the validity of a manual-calculation-friendly variation of the Legendre's formula: $\nu_p(n!)=\sum_{i=1}^\infty\lfloor\frac{n}{p^i}\rfloor$.

The Legendre's formula gives $\alpha$ in $$p^\alpha || n!$$ where $p$ is a prime number. To calculate $\nu_p(n!)$ on paper, one should normally find the quotients $q_i$ in these equations by long ...

### polynomial root of periodic continued fraction

Let $\xi \in \mathbb{R}$ be a periodic continued fraction expansion with $\xi =[\overline{c_0,\dots,c_n}], c_0 \neq0$ and $\frac{ k_i}{l_i}$ the approximate fraction with $1 \leqslant i \leqslant n$...

### 4 Number Theory from 2015 Yakutia Math Olympiad

Tuymaada 2015, Day 2, Problem 6, Senior League states: Consider integers $a,b,c,d$ such that $0 \leq b \leq c \leq d \leq a$ and $a>14$. Prove that there exists a positive integer $n$ that can not ...

### 4 Set of positive integers with unique sums

1 answers, 1.151 views combinatorics elementary-number-theory
What I'm looking for is the name of a type of number set. Given a number T (for total) and a set of positive integers S, I want to uniquely identify the subset of S that sums to T. All sets containing ...

### 3 Variation of a Beatty Sequence

I've been studying Beatty sequences lately, and, having read and understood a proof of Raleigh's theorem, I know that if $\lfloor \alpha x\rfloor$ is a Beatty sequence with $\alpha\gt 1$, then its ...

### Redundancy in solutions to a Diophantine equation.

Here, they write the solution with both (4d,d) and (12d,3d) and similarly (d,d) and (5d,5d). Aren't these two sets redundant as the second can be expressed as the first as d varies over the integers?...

### 3 Proving that there doesn't exist $n,m\in\mathbb{N}$ that satisfy $\phi(n)=2*13^{2m-1}$

I have to prove that there doesn't exist $n,m\in\mathbb{N}$ that satisfy $\phi(n)=2*13^{2m-1}$. My first idea is to think of primes in this situation: to investigate: $$\phi(3),\phi(13^{2m-1}+1).$$ ...

### Greatest common divisor property: $\gcd(a, b)=\gcd(a+nb, b)$

how to prove that $\gcd(a, b)=\gcd(a+nb, b)$ for any integer $n$? I tried to how that $\gcd(a, b) \leq \gcd(a+nb, b)$ and $\gcd(a, b) \geq \gcd(a+nb, b)$ but I ended up not being able to prove the ...

### 1 Is $a+b=ab=a^{'}+b^{'}=a^{'}b^{'}$ possible for different elements of $\mathbb Z / n \mathbb Z$?

Let $a, b, a^{'}, b^{'} \in \mathbb Z / n \mathbb Z\$such that $a+b=ab=a^{'}+b^{'}=a^{'}b^{'}$. Is it possible if $a^{'} \ne a\$ and $a^{'} \ne b$? I try to solve this problem for a couple of hours, ...

### 5 Do any (non-trivial) 2-chains of Pythagorean triplets exist?

Define an "integer 3-relationship" as a function $f(a,b,c)$ of three integer variables, together with the condition that this function must equal zero. Two examples would be the "Pythagorean triplet ...

### 3 What are the factors of the “Fermat Remainder”?

$\forall a,b \in \mathbb{Z}, p\in \mathbb{P}$, let $$F_p(a,b) = \frac{(a+b)^p-a^p-b^p}{p}$$ Note: $F_3 = ab(a+b)$ $F_5 = ab(a+b)(a^2+ab+b^2)$ $F_7 = ab(a+b)(a^2+ab+b^2)^2$ According to data ...

### 1 Solve $n!=(n+1)^k-1$ for $n,k$

Find all integer $(n,k)$ such that $n!=(n+1)^k-1$ My Attempt . Let $(n+1)=p$. Therefore,$L.H.S. \equiv {-1}\pmod p$ and $R.H.S. \equiv {-1}\pmod p$. So,$k=log_{p} ^{(p-1)!+1}$. ...