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

23.643 elementary-number-theory questions.

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 ...

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 ...

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 $1 \leqslant i \leqslant n$...

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 ...

Is this the correct way to convert fractions to duodecimals and duodecimals to fractions? Write $\left(\frac{7}{13}\right)_{12}$ as a duodecimal. $\left (\frac{7}{13}\right)_{12}=\left(\frac{7}{15}\...

If asked to express the generic pythagorean triplets satisfying $a^2+b^2=c^2$ you would answer $$ a = k(r^2 - s^2), b = 2krs, c = k(r^2+s^2) $$ with $k,r,s \in \Bbb N$ and $r>s$ and $r \ne s \...

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 ...

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 ...

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?...

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: $$\phi(3),\phi(13^{2m-1}+1).$$ ...

For the induction case, we should show that $(n+1)^4-4(n+1)^2$ is also divisible by 3 assuming that 3 divides $n^4-4n^2$. So, $$ \begin{align} (n+1)^4-4(n+1)&=(n^4+4n^3+6n^2+4n+1)-4(n^2+2n+1) \\ &...

I don't see how they got that $b-c \equiv0$ mod $(3)$ and $a-c \equiv0$ mod $(3)$? $\phantom{}$ ** Solution **

i tried this code in python , but this gives me only the possibles of 0,0,0 ? is my approach correct ? are there any other digit combinations that satisfy the expression? ...

I want to show that the last digit of $x^{n}$ in base $12$ when $n=2, 3, \ldots$ is $0$ when $x=6$ $4$ when $x=A$, $(A=10)$ and also the last digit is $x$ when $x=3,5,7,8,11$ and $n$ is odd. ...

Pick four integers $a,b,c$ and $d$. Then we get a corresponding sequence given by $$t_{n+2} = at_{n+1} +bt_n, \; t_1 = c, \;t_2 = d.$$ From what I can tell, we seem to get an especially rich theory ...

I had this problem. I solved it. tell me if it is correct I have to prove $6^{n+1} +7^{2n+1}$ is divisible by $43$ when $n\geq 1$ My solution $6^{n+1} +7^{2n+1}$=$216.6^{n-1}+343.49^{n-1}$ $\equiv 1....

I am familiar with the solution of "how to find all rational points on $x^2+y^2=1$ ?". I would like to know if i can solve: "how to find all rational points on $x^2+7y^2=1$ ?" using the same ...

I am reading Sierpinski's book: Elementary theory of number, 1988. In that book has the following problem (page 254): Problem. Prove that for arbitrary natural numbers $a,b$ there exist infinitely ...

Consider an urn which contains three different kinds of balls A, B and C. We suppose that there is at least one ball of each kind in the urn. We define the event $A$ as "to get, in $n$ independent ...

Let $$S \subset N_{100}= \{1,2,3,4, \dots, 100 \}$$ with $|S|=51$. Then prove for any such $S$ it is possible to select two numbers from $S$ such that one is a multiple of other. I wasn't able to ...

Dirichlet's theorem says that for any two positive coprime integers $a$ and $d$, the arithmetic progression $a,a+d,a+2d,a+3d,\ldots$ contains infinitely many prime numbers. In other word, there are ...

If $b \ge a$ and p is prime, then $(b-a)^p \equiv b^p-a^p \mod{p}$. I know this: $b^p = (b - a+a)^p \equiv (b-a)^p +a^p \mod{p} $. After all help in this comments I wrote this, I want to know if ...

Let $2 \leq p < q$ be two prime numbers. Consider $p+1$ and $q+1$. Observe that $d:=\gcd(p+1,q+1)$ can be 'anything': (a) If $(p,q)=(2,3)$, then $(p+1,q+1)=(3,4)$, so $d=1$: $p+1,q+1$ are ...

I am trying to prove that $(m+1)^3+(m+2)^3+\cdots + (2m)^3$ can never be square. $m$ is a positive integer. I have managed to show this but it was long and not very elegant. I am interested in seeing ...

So I have to prove: Let $p\ $ be a odd number, $\alpha\ $ natural number and r primitive root of $p^{\alpha}$, so $\ o_{p^{\alpha}}r=\phi(p^{\alpha})$ $r+p^{\alpha}$ is a primitive root of $2p^{\...

Find all solutions to $1!\times 3! \times \cdots \times (2n -1)!=m!$ I have found $(n,m)=(1,1),(2,3),(3,6),(4,10)$ How can one prove there are no more? Also is it a coincidence that these are the ...

So let r be primitive root of an odd prime p. I have to prove: $$o_{p^2}(r) \in \{ p-1, p (p-1) \}$$ So I know: $$o_{p^2}(r)= x , r^x\equiv 1 (mod \ p^2)$$ where x is the smallest such. But I ...

I'm reading Fermat's Theorem (that $a^{p-1} \equiv 1 \pmod{p}$) proof (What is mathematics book) and they consider the multiples of a $m_1 = a, m_2 = 2a, m_3 = 3a, ..., m_{p-1} = (p-1)a$. They explain ...

So I have this equality to prove for any $n \in \mathbb{N}$: $$ \sum_{d|n} \sigma(d) \phi(\frac{n}{d}) = n \tau(n) $$ So I was able to show that left and right side are multiplicative. So how can I ...

Prove that the fraction $\dfrac{n^3 + 2n}{n^4 + 3n^2 + 1}$ is in lowest terms for every positive integer $n$. I just don't know how to solve this. I tried polynomial division, expressing the gdc of ...

This question has been explored thoroughly, and in more generality too. For general fields, I am aware of standard proofs. However, I was naively trying to prove it in the simple case of prime $p$ ...

Given numbers $k$ and $n$ how can I find the maximum $x$ where: $n! \equiv\ 0 \pmod{k^x}$? I tried to compute $n!$ and then make binary search over some range $[0,1000]$ for example compute $k^{500}$...

I'm currently working through Don Bernard Zagier - Zetafunktionen und quadratische K├Ârper (1981) (German for: Zeta functions and quadratic fields). $K = \mathbb{Q}(\sqrt{d})$ is a quadratic ...

Does anyone know of any resources on questions on primitive roots and order of a modulo n? They need to be suitable for elementary number theory course. (These could be interesting results and ...

Suppose $a$ is a positive integer. When do the totient values of $a$ , $a+1$ and $a+2$ form a pythagorean triple ? In other words : For which positive integers $a$ does the equation $$\phi(a)^2+\...

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 ...

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, ...

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 ...

$\forall a,b \in \mathbb{Z}, p\in \mathbb{P}$, 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 ...

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,$k=log_{p} ^{(p-1)!+1}$. ...

I am hoping those with more experience in number theory than me can help me out on this question. After learning that $\gcd(a^m - 1, a^n - 1) = a^{\gcd(m, n)} - 1$, I questioned what is $\gcd(a^m + 1,...

Can some explain how the highlighted implication is so obvious i cant seem to get there from previous step

Let $q, q'\in\mathbb{N}$ and suppose that $q'$ divides $q$. Let $U(m)$ denote the multiplicative group of residue classes coprime to $m,$ that is $$U(m)=\left(\mathbb{Z}/ m\mathbb{Z}\right)^*.$$ ...

Question: Show that if $n$ is a positive integer such that $24$ divides into $n + 1$, then $24$ divides the sum of all divisors of $n$ (denoted in number theory by $\sigma(n)$ or $\sigma_1(n)$). ...

Proof $3^{2n}-2^n$ is multiply of 7 First $n=1$, $9-2=7$ For N+1, $3^{2(n+1)}-2^{n+1}=3^{2n+2}-2^n\cdot2$=$9\cdot3^{2n}-2\cdot2^n$ Can someone give me hint how can i get multiply of 7?

I have been dealing with modular arithmetic lately but I am stuck in huge exponents. I know how to calculate easily up to some exponents. But for example an exercise like : $$3^{3000} \pmod{15}\\6^{...

How do you find the divisors of $3^{32} - 1$ between $75$ and $85$ without using a computer?

Let $n$ be a positive integer. Assume that: $N_k$ is the number of pairs $(a,b)$ of non-negative integers such that $ka+(k+1)b=n+1-k$. Find $N_1+N_2+\cdots N_{n+1}$. I was trying to solve this ...

Wikipedia says something surprising: if we are using quadratic hashing, we can alternate the sign of the offset to avoid collisions. I can't find any counterexamples, but I can't prove it, either. I'...

I was working through a paper on efficient congruencing when I came across the following, but I can't understand why it is correct. For reference $\xi$ is an integer and $1 \leq \xi \leq p$ where $p$ ...

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