# Tagged Questions

Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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### Collatz conjecture varient prove

Every body knows that Collatz conjecture cannot be proved.It works like this: 1.For any odd number $n$ it gives $3n+1$. 2.For any even number $n$ It gives $\frac{n}{2}$. Now the problem is open ...
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### If $a,b,c>0$, then can we find values such that the given condition is valid?

For what integer values of $a, b, c$ the expression $(1-a)(1-b)(1-c) = abc$ ?
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### The $25$th digit of $100!$

I want to find The $25$th digit of $100!$. My attempt:It is easy to know it has $24$ zeroes.Because: $\lfloor {\frac{100}{5}} \rfloor+\lfloor {\frac{100}{25}} \rfloor =24$ By getting the fist ...
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### Have I found all the numbers less than 50,000 with exactly 11 divisors?

The math problem I am trying to solve is to find all positive integers that meet these two conditions: have exactly 11 divisors are less than 50,000 My starting point is a number with exactly 11 ...
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### Are all theorems usable? [closed]

The (revised) question to answer: Can anyone give an example of a serious proof using this funny (revised) theorem? For any natural number $n$ and prime $p<n-1$ there exist a prime $q$ ...
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### There exists infinitely many $n\in\mathbb{Z}$ such that $f(n)$ is a prime.

I found in a number theory book the following lines Let $f(x)$ be a non-constant polynomial with integer coefficients such that none of the following hold for it 1) There is an integer $d>1$ ...
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### The greatest common divisor of $(O_n, T_n+2)$ where $O_n$ and $T_n$ are the oblong and triangular numbers respectively.

Suppose that $T_n$ is odd. Can we find infinitely many $n$ such that $(O_n, T_n+2)=1$? Is it trivial and obvious? My hunch based on some hand calculations is to look at $n$ congruent to $0$ or $2$ ...
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### Prove that there are exactly $k$ pairs $(x,y)$ of rational numbers with $0\leq x,y<1$ for which both $ax+by,cx+dy$ are integers.

Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x,y)$ of rational numbers with $0\leq x,y<1$ for which both $ax+by,cx+dy$ are ...
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### Arithmetic mean of the integers in set $S=\{k:k\in \mathbb{Z}, 1\leq k\leq n$ and $gcd(k,n)=1\}$

Or stated simply, what is the arithmetic mean of the totatives of $n$? From this question here I can see that the sum of the totatives is given by the formula $\large\frac{n\times\phi (n)}{2}\large$. ...
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### How does $x^4+y^4=z^2 \implies x^4+y^4=z^4$?

Why is the statement "the following cannot be satisfied" for $x^4+y^4=z^2$ more strong than for $x^4+y^4=z^4?$ More specifically, how does $x^4+y^4=z^2 \implies x^4+y^4=z^4?$ This statement was ...
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### Relation between HCF, LCM and product of multiple numbers [duplicate]

It is well known that for two numbers $a$ and $b$, $$\text {lcm} (a,b)\times \text {hcf} (a,b)=ab$$ Does there exist a similar equality/ inequality between HCF, LCM and product of multiple numbers? (...
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### Prove that $\dfrac{b^{n-1}a(a+b)(a+2b)\cdots(a+(n-1)b)}{n!}$ is an integer

Let $a$ and $b$ be integers and $n$ a positive integer. Prove that $$\dfrac{b^{n-1}a(a+b)(a+2b)\cdots(a+(n-1)b)}{n!}$$ is an integer. Define $v_p(x)$ such that if $v_p(x) = n$, then $p^n \mid x$ but ...
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### Help on an application of Dirichlet's theorem for primes in progression

Suppose that I have an infinite sequence of positive integers $$a_1,\ldots,a_m,\ldots$$ with the following recursion $$a_{m+1} -a_m =b(m+1)$$ So that $$a_{m+1} =b(m+1) +a_m$$ Suppose ...
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### Steep Diagonals and Magic Squares

We want to describe via a picture a set of subsets of a square which are something like diagonals, but are not quite the same. We’ll call them steep diagonals. One of them, labelled e, is illustrated ...
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### Positive integer solution $pm = qn+1$

Let $m,n$ be relatively prime positive integers. Prove that there exist positive integers $p,q$ such that $pm = qn+1$. We know Bézout's identity that there exist integers $p,q$ such that $pm+qn = 1$,...
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### Solving a system of congruences with unkown moduli?

So I have two congruences of the form: n[1]=r[1] mod m and n[2]=r[2] mod (a*m+b) with known n,r,a, and b. Is there a way way to efficiently get (an acceptable) m? Edit: I mean is there any way ...
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### How many integer solutions for $a = n(4m-1)/4b$?

For the equation $$a = \frac{n(4m-1)}{4b}$$ where $n,m,a$ and $b$ are positive integers and $1\leq a,b\leq n$, how many valid, unique solutions $(a,m)$ exist for fixed $n$ ...
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### Primes in the binomial transform of $[1, 1, 2, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, …]$.

This question is related to this sequence A139482. A commentator gives the following formula for $a_m$ $$a_m = {3m^2-9m+10 \above 1.5pt 2}$$ I have that you should consider the sequence $b_n =3n+2$ ...
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### Find all even numbers that can be represented as a difference of squares in only two ways

I am currently working on this proof. I am looking to find (with proof) all even numbers that can be represented as a difference of squares in only two ways. My thoughts thus far. I examined the ...
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### Let $a,b$ be positive integers that are coprime, then $ac+1=bd$ for some positive integers $c,d$.

Prove that: Let $a,b$ be positive integers that are coprime, then $ac+1=bd$ for some positive integers $c,d$. I know that by Bezout's identity, there exists integers $c',d'$ such that $ac'+bd'=1$...
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### Find all integer roots of: $x^2(y-1)+y^2(x-1)=1$

Find all integer roots of: $x^2(y-1)+y^2(x-1)=1$ Obviously $(2,1)$ and $(1,2)$ are two answers. But I was unable to manipulate the equation algebraically giving a useful form for finding all other ...
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### Proving there is a unique binary operation we call multiplication

The following is a theorem from the book The Real Numbers and Real Analysis by Bloch which I am currently self-studying. I am pretty sure my proof for uniqueness is correct, but I am wondering is ...
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### What's the definition of a prime Gaussian integer? [closed]

The units are considered to be primes, namely, $1,-1,i,-i$. Why?
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### Conjecture about divisibility: if $d \mid n$, then there exists $r,s$ such that $n=r+s$ and $d = \gcd(r,s)$

Given $n\in\mathbb Z^+$. If $d<n>1$ and $d\mid n$ it exists $r,s\in \mathbb Z^+$ such that $n=r+s$ and $d=\gcd(r,s)$.
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### Need help with the proof of a theorem about Gaussian integers

Theorem 6-3. If $\alpha$ and $\beta$ are integers of $Z[i]$, and $\beta \neq 0$ then there are $\kappa$ and $\rho$ in $Z[i]$ such that $$\alpha =\beta\kappa+\rho, \text{ } N_\rho < N_\beta$$ ...
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### How can be proven that any number X is greater,lesser or equal to any other number Y?

I have looked for it on the internet, really, but all I have found are particular cases like 1 > 0, or such. Is there an algebraic proof for proving that x > y or, x = y, or x < y? I thought of ...
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### If $L > 1$ is an odd almost perfect number with $\omega(L)=6$, then $L$ must be divisible by $3$.

Edited July 15 2016 Let $\mathbb{N}$ denote the set of positive integers. Let $\sigma = \sigma_{1}$ denote the (classical) sum-of-divisors function. Let $I(x) = \dfrac{\sigma(x)}{x}$ denote the ...
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### Smallest chain of consecutive integers not all coprime

Let $t$ be a positive integer. What is the smallest $t$ for which we can find an integer $a$ such that each element of the set $\{a+1,a+2,\dots ,a+t\}$ is not coprime with all other elements of the ...
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### lower bound for sum of distinct n-th roots of unity

Given a positive integer $n$, define $\zeta = e^{2\pi i/n}$ and define $s: \mathbb Z^n \to \mathbb C$$s(\vec x) = \sum_{k=0}^{n-1} x_k \zeta^k$$ Let us consider the set$S = \{ |s(\vec x)| : \vec x \...
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### Can it proved that the GCD does not divide the integer coefficients in the linear form of the GCD?

Let $d = (a,b)$ then $d = ax +by$ for some $x,y \in \mathbb{Z}$ I want to prove that $d \nmid x,y$. Motivation I'm trying to solve the following problem: If $a$ is prime to $b$ and $y$, $b$ is ...
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### Showing that $2^6$ divides $3^{2264}-3^{104}$

Show that $3^{2264}-3^{104}$ is divisible by $2^6$. My attempt: Let $n=2263$. Since $a^{\phi(n)}\equiv 1 \pmod n$ and $$\phi(n)=(31-1)(73-1)=2264 -104$$ we conclude that $3^{2264}-3^{104}$ is ...
How to prove that following hypothesis is true ? Definition Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ , where $m$ and $x$ are ...
### Prove that the modular congruence holds: $b^d$ $=$ $r \pmod n$, $b^{d/q}$ $=$ $x \pmod n$, then $x^q$ $=$ $r \pmod n$.
Prove that if $b^d$ $=$ $r \pmod n$ $b^{d/q}$ $=$ $x \pmod n$, then $x^q$ $=$ $r \pmod n$ for any integers $b$, $n$, $r$, and $q$ (which divides $d$). Or more simply that $b^d$ $=$ $x^q$ \$\...