# Tagged Questions

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

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### Need help with the explanation of a theorem

http://people.ucsc.edu/~yorik/Math110/PDF/QuadRec.pdf My question is in theorem 3. I understand until it says $a={p-1}/2$ if $p \equiv 1(\mod 4)$ and $a=p-(p-1)/2$ if $p \equiv 3(\mod 4)$. Why is ...
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### Prove that Carmichael number has no primitive roots

Prove that if $n$ is a Carmichael number, then $n$ has no primitive roots. This seems tricky to prove, and the only logical explanation for this is that it contradicts the basis of the Lucas Primality ...
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### There are infinitely many odd numbers not expressible as the sum of a prime number and a power of $2$

Prove that there are infinitely many odd integers that are not expressible as the sum of a prime number and a power of two. This is a difficult problem. Please give me some hints and some examples of ...
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### Is it possible that $(\text{even}/\text{even})\times \text{even} = \text{odd}$? [closed]

Do even natural numbers $e_1,e_2,e_3$ exist with $\frac{e_1}{e_2}\cdot e_3=o$, such that $o$ is an odd natural number? Find a relation for these three even numbers $(e_1,e_2,e_3)$.
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### Theorem on arithmetic of natural numbers.

From "Analysis I"-Herbert & Joachim: (starting from the Peano axioms) "There are operations addition + , multiplication · and a partial order ≤ on N which are uniquely determined by the ...
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### Solve $a\equiv (105^{70}+1)^{15}\pmod {51}$

Find a number $a\in[0,50]$ such that $a\equiv (105^{70}+1)^{15}\pmod {51}$ My question is there a simple way to solve this by hand? maybe using Fermat's little theorem or something?, I tried this ...
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### The Greatest Common Divisor of All Numbers of the Form $n^a-n^b$

For fixed nonnegative integers $a$ and $b$ such that $a>b$, let $$g(a,b):=\underset{n\in\mathbb{Z}}{\gcd}\,\left(n^a-n^b\right)\,.$$ Here, $0^0$ is defined to be $1$. (Technically, we can also ...
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### What is this operator called?

If $x \cdot 2 = x + x$ and $x \cdot 3 = x + x + x$ and $x^2 = x \cdot x$ and $x^3 = x \cdot x \cdot x$ Is there an operator $\oplus$ such that: $x \oplus 2 = x^x$ and $x \oplus 3 = {x^{x^x}}$? ...
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### Are there any two identical terms in this series, defined parallely to the primes? [closed]

Let $p_n$ denote $n$-th prime number and $k_n$ be sequence that is \begin{align} k_1 &= 1 \\ k_2 &= p_2 - k_1 &&( 3-1 = 2 ) \\ k_n &= p_n - k_{n-1} &&\text{( n is integer ...
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### Question about Lagrange's and factor theorem

Find all solutions $x^{12} \equiv 1 \pmod{13}$. Hint: the computation of high powers is better accomplished by using binary expansion of the exponent. Approach: This time I don't have a clear ...
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### Number Theory GCD/LCM Problem

The following is from a problem set: Let $a,b,c$ be three positive integers such that $$\text{lcm}(a,b) \cdot \text{lcm}(b,c) \cdot \text{lcm}(c,a) = a \cdot b \cdot c \cdot \gcd(a,b,c).$$ ...
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### Show that for all $a\in\mathbb{N}$, there exists $b\in\mathbb{N}$ and square-free integer $c$ such that $\sqrt{a}=b\sqrt{c}$.

I'm having some difficulties continuing this problem. I get that $b^2\mid a$ and $c\mid a$ but I am not sure where to go from there.
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### Solution to Divisibility Problem

I have attempted to solve problem which is stated as follows: $2^n+1=xy$ where $n>0$ and $x,y>1$. Prove that $2^a$ divides $x-1$ iff $2^a$ divides $y-1$. My solution is as follows: $x$ ...
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### Prove that $a \mid k$ if $a \mid k·c$, $a \mid k·b$ and $gcd(c,b)=1$ for all $a,b,c,k \in \mathbb{Z}$ [closed]

Let $a,b,c,k \in \mathbb{Z}$ and $a \mid k·c$, $a \mid k·b$ and $gcd(c,b)=1$. Prove that $a \mid k$.
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### A more sophisticated method of finding all $n$ satisfying $n! = (k + 1)(k-1)$?

Find all $n$ such that $n!$ can be expressed in the form $(k+1)(k-1)$ where $k$ is an integer $>1$. This problem stems from another problem that I tried to solve from David Burton's number theory ...
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### Question about oblique co-ordinate system

Definitions: Let the following figure show an oblique $2$ dimensional co-ordinate system, where $O$ is the origin and the parallelogram $OQRP$ is called the fundamental parallelogram. Rest of the ...
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### Hypothetical equation (modulo a power of two) and the value [duplicate]

We have hypothetical equation: $2^{b} \% k = z$. Assume that we know $z$, $b$ and $k$. So everything! We want to know only if the above equation is true. I do not want to use the exponentiation ...
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### Existence of the natural density of the strictly-increasing sequence of positive integer?

Let $A=\{a_n\}$ is a strictly-increasing sequence of positive integer. The natural density of this sequence is defined by $\delta(A)=\lim_{n\rightarrow \infty} \frac{A(n)}{n}$ whenever the limit ...
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### How to provide Mathematical Proof for number theory scheme?

I have a set S={1,2,...,N-1}. N=pq (where p and q are RSA prime numbers). Scenario is that User need to retrieve the Database blocks without revealing his block index to the Server i.e, Private ...
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### Checking proof of simple number theory problem

I came up with a solution to a number theory problem. Please check it. Prove that $a^2 + b^2 + c^2 + d^2$ is never a prime if $ad=bc$, where $a,b,c,d$ are positive integers. We will prove the more ...
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### Proof about Lagrange's theorem in number theory

Based on the textbook Lagrange's theorem states: The congruence $$f(x) \equiv 0\pmod p$$ in which $$f(x)=a_0x^n+....+a_n,\text{ } a_0\not\equiv0\pmod p$$ has at most $n$ roots. $p$ is a prime number ...
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### Properties of a finite field extension of degree 2.

I am bad (but trying to improve!) at very basic number theory and algebra. I'm quite sure this question is easy, but I do not know what fundamentals I am missing. This is from Ireland & Rosen's "...
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### Question about Lagrange's theorem

Based on the textbook Lagrange's theorem states: The congruence $$f(x) \equiv 0(mod\text{ } p)$$ in which $$f(x)=a_0x^n+....+a_n,\text{ } a_0\not\equiv0(mod\text{ } p)$$ has at most n roots. p is a ...
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### Question about polynomial congruences

$x^3 \equiv x(mod$ $3)$ for all x, whereas obviously $x^3$ and $x$ are not algebraically congruent $(mod$ $3)$. What does it mean to be algebraically congruent?. In this case the two polynomials can ...
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### Induction Implies Well Ordering

Every proof that induction implies well ordering I have seen goes: assume $S\subset\mathbb{N}$ has no least element and let $T$ be its complement with respect to $\mathbb{N}$. Since $1$ is the ...
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### General solution of Pell's equation

If we know the minimal solution or any specific solution of Pell's equation $x^2-ny^2=1$ , is there is any general formula to write all solution of Pell's equation?
### Prove that we cannot found any simple algebraic sloution for $n>2$
for this sequance we can write this formula:$\{1-1,1,-1,1,-1,1,...\}=(-1)^n$ for this also we can do that: $\{-1,-1,1,1,-1,-1,1,1,-1,-1,...\}=(-1)^\frac{n(n+1)}{2}$ But how to prone there is no ...