# Tagged Questions

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

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### Does there is a converse for this result, i.e., $Something$⇒ $p$ is a prime

One of the known results in number theory is the following: If $p$ is a prime and if $a$ is any integer, then $$a^{p}\equiv a\pmod{p}$$ My question is: Does there is a converse for this result, i.e.,...
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### Is there a difference in the rate of decrease between $f(x)$ and $g(x)$ for increasing $x$?

I have the following two functions of $x$: $f(x) = \frac{c}{c + (N-1)o + Nd + xl}$ $g(x) = ae + (1-a)\frac{1}{x+2N}$ with $0 \leq a, e, c, o, d, l \leq 1$ and $N, x \in \mathbb{N}^+$. For both ...
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### Why can't prime numbers satisfy the Pythagoras Theorem? That is, why can't a set of 3 prime numbers be a Pythagorean triplet?

Suppose $a$, $b$ and $c$ are three prime numbers. How to prove that $a^2 + b^2 \neq c^2$?
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### Find all primes $p$ such that $z$ is also a prime number

Let $p$ be a prime number. We know that $z=(-√3+2)^{2^{p-2}}+(√3+2)^{2^{p-2}}$ is an integer. My question is: Find all primes $p$ such that $z$ is also a prime number.
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### solve this simple equation:$ax^2+byx+c=0$

I need help solving the diophantine equation:$$ax^2+bxy+c=0$$ The quadratic formula does not seem to help much. Please help.
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### Let $\{A_n\}_{n=1}^\infty$ be a sequence of nonempty subsets of $\mathbb{Z}$, which of the following is uncountably infinite?

Let $\{A_n\}_{n=1}^\infty$ be a sequence of nonempty subsets of $\mathbb{Z}$. Which of the following is uncountably infinite? A) $A_1$ B) $\bigcap_{n=1}^\infty A_n$ C) $\bigcup_{n=1}^\infty A_n$ D)...
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### Prove that $(\sqrt3+2)^m$ is not a natural number for all natural numbers $m≥1$

The aim of this question is to show this lemma: Prove that $(√3+2)^{m}$ is not a natural number for all natural numbers $m≥1$.
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### Use Induction to prove: $(1+2x)^n \geq 1+2nx$

Show by induction that: for all $x>0$ that $(1+2x)^n \geq 1+2nx$ So far I have: for $n=1 \rightarrow (1+2x)^1 \geq 1+2x$. True! for $n=k+1 \rightarrow (1+2x)^{k+1} \geq 1+2(k+1)x$ ...
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### Discrete Math Proof Method

Give a direct proof of the fact that $a^2-5a+6$ is even for any integer $a$. Suppose $a$ and $b$ are integers and $a^2-5b$ is even. Prove that $b^2-5a$ is even.
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### Largest prime number with all digits different

What is the largest prime with distinct digits? (It is certainly less than ten digits long.Can you explain it why?
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### Prerequisites: Dirichlet Lectures Number Theory

I am interested in getting Dirichlet's Lectures in Number Theory but I'm afraid I don't know that much advanced math. Do I need to know things like Determinants for this book? Any list of ...
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### Number of $0$ in the end of $11^n-1$

$n$ is integer, calculate number of $0$ in the end of $11^n-1$(i.e. largest integer $m$ such that $10^m|11^n-1$). The original question was $n=100$ and I could only choose $m$ from 1 to 5. I ...
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### Prove that exist set $B$: $|B|\ge 2014$ .

For $A$ is a set has $2014$ natural numbers. Prove that exist a set $B\subset \mathbb{N}$ such that $A\subset B$ and sum square of all elements of $B$ equal area of all elements of $B$. I think we ...
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### The sum of n numbers that cube to one is congruent modulus three.

Assume $a_1,\dots,a_n\in\mathbb C$ cube to give one. Assume $\sum a_i=\sum a_i^2$. How can we see that $\sum a_i\equiv n(mod3)$? May the sum be different than $n$?
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### Using Hensel's lemma to solve congruence?

I'm trying to use Hensel's lemma to solve the congruence $$x^3 + x^2 - 5 \equiv 0 \pmod{7^3}$$ I begin by solving $$x^3 + x^2 - 5 \equiv 0 \pmod{7}$$ and observe that $x \equiv 2 \pmod{7}$ is the ...