Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc.. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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1answer
35 views

Prime $4n+3$ simple proof?

Let $p=4n+3$ be a prime. Prove that $\prod_{k=1}^{p-1}(x+k^2)\equiv (x^{\frac{p-1}{2}}+1)^2\pmod p$. Is there a simple proof that doesn't use say arithmetic in $\mathbb{Z}[i]$? My approach was to ...
0
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0answers
9 views

The Nagell-Ljunngren Equation

I have been trying to find the papers of Nagell and Ljunngren, which deal with the equation (x^n-1/x-1=y^2) and solve it completely.Many papers cite these papers, but I haven't found them anywhere.If ...
0
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2answers
37 views

Find all prime numbers $p$ such that $p \mid 2^p + 1$

I know that they somehow look like Mersenne primes $2^p-1$ but in this case we have $2^p+1$. Here is my attempt. If $p \mid 2^p+1$ then $ \exists k \in Z$ such that $pk = 2^p+1$ or that $2^p \equiv ...
0
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1answer
28 views

Transforming quadratic forms, how is this theorem called?

In my textbook there is the following nameless theorem: Let $Q=\sum_{i,j=1}^n a_{ij}X_i X_j$ with $a_{ij}=a_{ji}\in K$ be a quadratic form in $n$ variables over a field $K$ not of characteristic ...
1
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1answer
19 views

What is the discriminant of $R:=\mathbb Z[\sqrt{2},\sqrt{3}]$?

What is the discriminant of $R:=\mathbb Z[\sqrt{2},\sqrt{3}]$ ? The discriminant is defined as the determinant of the matrix $\left(tr(x_ix_j)\right)_{1\le i,j\le n}$ for any basis ...
1
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0answers
9 views

Small proximity of important points of a function

Let a,b,c be coprime integers with a^2 + b^2 > c^2 and consider the function f(x) = a^x + b^x – c^x. It is easy to verify that there exist r and s such that f(r) ≥ f(x) for all x and f(s) = 0. Prove ...
1
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0answers
14 views

On the existence of a polynomial in several variables with only one zero

Given an ordered field $\mathbb{K}$, then for all $n>1$ there exists $f\in\mathbb{K}[X_1,\ldots,X_n]\ s.t.\ \mathcal{V}^{\mathbb{K}^n}(f):=\{p\in\mathbb{K}^n:f(p)=0\}=\{(0,\ldots,0)\}$ For ...
3
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3answers
85 views

Prove there exists $m > 2010$ such that $f(m)$ is not prime

Let $$f(x) = \sum_{i = 0}^n a_ix^i$$ be a polynomial with $a_i \in \mathbb Z, n > 0, a_n \neq 0$. Prove that there exists some natural number $m>2010$ such that $|f(m)|$ is not a prime number. ...
2
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2answers
44 views

“$111 \dots$ upto $3^n$ digits” is divisible by $3^n$

Prove that an integer of the form "$111 \dots$ upto $3^n$ digits" is divisible by $3^n$ My attempt For $n=1,$ $111$ is divisible by 3. Let $T_n=111...$ upto $3^n$ digits is divisible by $3^n$. ...
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2answers
46 views

Giving integer images [on hold]

Prove the question below for all distinct positive integer X, Y, Z and n. See, please, Around Fermat Last Theorem where the question has been misunderstood.
2
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1answer
16 views

Finding the maximal order in a number field

Finding the maximal order in a number field Suppose the number field is $K:=\mathbb Q(\alpha)$, then $\mathbb Z[\alpha]$ is not necessarily the integral closure of $\mathbb Z$ in $K$. I know the ...
0
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1answer
23 views

A question in Number Theory - prove there exist m>2010 s.t f(m) is not prime

Let $$f(x)=\sum_{i=0}^n a_nx^n$$ be a polynomial with $$a_n \in Z,n>0,a_n\neq0$$ Prove that there exists some natural number $$m>2010$$ such that $$|f(m)|$$ is not a prime number. I tried to ...
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0answers
60 views

Around Fermat Last Theorem [on hold]

HINT.- According to an old and still unproven conjecture of V. Bouniakowsky, P(n) is infinitely often a rational prime. Here we need just one prime to arrive to the conclusion. SKETCH ...
0
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1answer
35 views

A question about primes, number theory [duplicate]

I tried to solve this question but without a success: Let $p$ be a prime number,and $p^2+2$ is also prime, prove that $p=3$. I tried to show $p^2+2$ as a product of numbers and then to show that ...
2
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1answer
19 views

Find smallest discrete logarithm, knowing some discrete logarithm.

Discrete logarithm is a value $x$ that satisfy the equality $$a^x \mod m = b$$ Is there an easy way to find the smallest possible discrete logarithm, knowing some other discrete log. Basically if I ...
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0answers
22 views

Hyperbolic curves and elliptic curves

I am so sorry, if I am very wrong. I know some what hyperbolic functions/curves, and elliptic curves as well. Now my question is that; 'Is there hyperbolic elliptic curves?. If yes, what are the ...
1
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1answer
12 views

Converting Unbalanced Ternary Numbers to Balanced Ternary Number

Can someone please provide a step by step algorithm for converting unbalanced ternary to balanced? for instance: (Base 10) 501 = (Base 3 Unbalanced) 200120 I've done some research on this conversion ...
0
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0answers
20 views

genus of an algebraic curve [on hold]

I would like to draw the better answer to my question and I believe that, math-stack will help me out. How and why to find GENUS of an algebraic curve? Is there any relation between genus and ...
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votes
0answers
45 views

some amazing properties of combinatorial numbers [on hold]

I want to prove $$ C_{2^{i+1}-k-1}^k=\frac{(2^{i+1}-k-1)(2^{i+1}-k-2)\cdots(2^{i+1}-k-(k-1))(2^{i+1}-2k)}{k(k-1)\cdots 2\cdot 1} $$ is even, for all $k=1,2,3,\cdots, 2^i-1$. Here $i\geq 1$. How to ...
2
votes
1answer
13 views

How are Digit Extraction Formulas Special?

There are hundreds of similar looking formulas to the BBP that I've seen on the internet, but those are termed as spigot algorithms only. Why is it that none of those other pi formulas can be used ...
0
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2answers
38 views

Prove that if $n \equiv 7 \pmod 8$, then $n$ cannot be expressed as the sum of three squares.

I begin by contradiction. Assume that $n$ can be expressed as the sum of three squares. That is $n = a^2 + b^2 + c^2$. Now since $n \equiv 7 \pmod 8$ then $8 \mid n - 7$ so $8 \mid a^2 + b^2 + c^2 - ...
3
votes
1answer
48 views

Negative Pell's Equation: Prove that $k=3$.

I made this problem (while solving another problem) but I haven't been able to prove it. Let $x,y,k\in \mathbb{Z}^+$. Prove that if $x^2-(k^2-4)y^2=-1$ then $k=3$. Any pointers are appreciated, but ...
7
votes
2answers
112 views

$\forall x\in\mathbb{R},,$ if $x^{2}$ is rational, then $x$ is rational.

This is my attempt at this question. Is this correct? $\forall x\in\mathbb{R},,$ if $x^{2}$ is rational, then $x$ is rational. This statement is false. Using counterexample, let $x=\sqrt{2}$. Since ...
3
votes
3answers
78 views

What is the remainder of $314^{164}$ divided by 165?

What is the remainder of $314^{164}$ divided by 165? Since 165 is not a prime, we cannot apply Fermat's Little Theorem directly. However since $165=3\times 5\times 11$, we could try to divide ...
1
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1answer
26 views

Does there exist a quadratic generalization of the continued fraction approximants?

Let $t$ be a real number and let $\frac{p_n}{q_n}$ be its continued fraction approximants. These have the property that $$ \left| t - \frac{p_n}{q_n} \right| < \frac{1}{q_n q_{n+1}} $$ In other ...
3
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0answers
25 views

Multiples of some set has density

Nathanson gives a proof in Elementary Methods in Number Theory (Theorem 7.14) that, if a set $S$ of positive integers has $$ \sum_{s\in S}\frac1s<+\infty $$ then the set of positive multiples of ...
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0answers
18 views

Is 1 a quatratic residue modulo any number? [on hold]

For any number n, is 1 always a quadratic residue mod n?
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0answers
66 views

prove that $x^2 + 5 =y^3$ has no solutions for $x,\ y \in \mathbb{Z}$ [duplicate]

So the question is completely stated by the title. My own thoughts: I can prove that $x^2 + 1 = y^3$ has no solutions for $x,y \in \mathbb{Z}$ by using the factorization: $$ y^3 = (x-i)(x+i) $$ in ...
0
votes
2answers
48 views

Integer solution to the equation

Does there exists an integer solution (for every integer $m\geq 1$) for the following equation? $$x_1x_2...x_n+(2y+1)z+y=4m+3$$ where, $1\leq x_1\leq x_2\leq...\leq x_n\leq l$,$0\leq y \leq ...
0
votes
3answers
15 views

system of conguences

I am trying to figure out how to solve: Find $x, y \in \mathbb{Z}$ such that $$2x+y\equiv 4\pmod{17}$$ and $$5x-5y\equiv 9\pmod{17}.$$ I know this means $2x+y-4$ and $5x-5y-9$ are multiples of $17$, ...
2
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1answer
17 views

Looking for different proofs for $-1$ is a quadratic residue of primes of the form $4k+1$ and related facts

Suppose $p$ is a prime of the form $4k+1$ , then $4|p-1=|\mathbb Z_p^*|$ , as $\mathbb Z^*_p$ is a cyclic group , so there is $\bar x \in \mathbb Z_p^*$ such that $o(\bar x)=4$ , then $o(\bar x^2)=2$ ...
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0answers
17 views

Question related to image of $[1,N]^n$ under a linear tranformation

I am reading an article and I am a bit confused about the following passage. I would appreciate any clarification. It goes as follows: Let $\bar{F}$ be a collection of $r$ linearly independent ...
0
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1answer
45 views

Does this compound interest problem coincide to the value of e by coincidence?

An account starts with €$1.00$ and pays $100\%$ interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be €$2.00$ . What happens if the ...
2
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1answer
44 views

$3 X^3 + 4 Y^3 + 5 Z^3$ has roots in all $\mathbb{Q}_p$ and $\mathbb{R}$ but not in $\mathbb{Q}$

This is an exercise in my textbook in a chapter about the Hasse-Minkowski-theorem: Show that the polynomial $3 X^3 + 4 Y^3 + 5 Z^3$ has a non-trivial root in $\mathbb{R}$ and all $\mathbb{Q}_p$. ...
0
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0answers
26 views

Determine all $n \in \mathbb{N}$ such that $GCD(n,48)=6$, $14|n$ and $|Div^+(n)|=12$.

Determine all $n \in \mathbb{N}$ such that $\gcd(n,48)=6$, $14|n$ and $|Div^+(n)|=12$. What I did: $14|n$ then $2|n$ and $7|n$ so $n=2\cdot7\cdot q$, $q \in \mathbb{Z}$. Then $6|n$ implies $2|n$ and ...
4
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0answers
37 views

Is there anything known about the value where the Euler and Hadamard products for $\zeta(s)$ are equal?

Take the Hadamard product for the Riemann $\xi$-function ($\rho$ is a non-trivial zero of $\zeta(s)$): $$\xi(s) =\frac12\, s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s) ...
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2answers
68 views

Squares numbers into squares

Solve the two related questions below in which lowercase letters are digits in base $10$, $a > 0$ and $N\in\Bbb N$. Find the values of $N$ in $(1)$ and prove or deny $(2):$ $$ ...
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0answers
22 views

the asymptotic approximation of a sum

$p_{n}$ and $p_{j}$ are two primes with $p_{n}<p_{j}$ where the $n$ and $j$ denotes the $n$th and the $j$th prime. I have this sum $$\sum \limits^{k=\frac{b-p^{2}_{n}p_{j}}{2p_{n}p_{j}} ...
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2answers
47 views

Are there arbitratily long runs of consecutive integers n that are NOT of the form $n = p^k$ or $n = 2p^k$ for some $k>0$ and $p$ an odd prime?

I know that there are arbitrarily long runs of consecutive non-primes. I am interested in this question because an integer $n>4$ has a primitive root if and only if it has the form $n = p^k$ or ...
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0answers
18 views

Generate a function that shuffles a number withing a given range which is reproducible

Lets say I have an array of numbers $1 2 3 4 5 6 7$. I want to shuffle these numbers in some order , $7 5 4 3 1 26$ . However , it should be revesrible. That is given the second array I must be able ...
0
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1answer
20 views

Exercise 2.13 from a computational introduction to number theory and algebra

This is an exercise from V. Shoup. A computational introduction to number theory and algebra. Let $p=2, e=3, a=b=1, c = 0$, then $p^{2e} = 64, z\in \{0,1,2,\cdots,63\}$, the conclusion is, there ...
1
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1answer
55 views

Show that $\limsup_{x \to \infty} \frac{\pi(x)}{x/ \log x} \geqslant 1. $

Show that $$\displaystyle\limsup_{x \to \infty} \dfrac{\pi(x)}{x/ \log x} \geqslant 1. $$ I've seen $\displaystyle\lim_{x \to \infty}$ operator, but I haven't seen $\displaystyle\limsup_{x \to ...
1
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2answers
47 views

What is the value of $1^2 + 2^2 + 3^2 + \cdots + (p-1)^2\pmod{p}$?

What is the value of $1^2 + 2^2 + 3^2 + \cdots + (p-1)^2\pmod{p}$? Let's try a several primes greater than 3... If $p=5$, then we have $1^2 + 2^2 + 3^2 + 4^2 = 30$, so that $30\pmod{5} = 0$ If ...
6
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2answers
79 views

Prove or disprove : $a^3\mid b^2 \Rightarrow a\mid b$

I think it's true, because I can't see counterexamples. Here's a proof that I am not sure of: Let $p_1,p_2,\ldots, p_n$ be the prime factors of $a$ or $b$ \begin{eqnarray} a&=& ...
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0answers
25 views

Prove that $a_{i,1}x_1 +a_{i,2}x_2 +···+a_{i,n}x_n ≤c_i, 1≤i≤n $ are all satisfied by a nonzero $n-tuple$ of integers.

My setting is that $c_1, · · · c_n$ are positive real numbers, and $A = [a_{i,j} ]$ is an $n × n$ non-singular matrix. Assume that $c_1 · · · c_n > | det(A)|.$ I want to prove that the n-linear ...
4
votes
2answers
54 views

Show that $\sum\limits_{p \leqslant x}1/p = \frac{\pi(x)}{x} + \int_2^x \frac{\pi(u)}{u^2} du.$

Show that $$\displaystyle\sum\limits_{p \leqslant x}1/p = \dfrac{\pi(x)}{x} + \int_2^x \dfrac{\pi(u)}{u^2} du.$$ In the equation above, $\pi(x)$ denotes the prime counting function. To get ...
1
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0answers
52 views
+50

Product of Stirling Numbers of the first kind

I have been messing around with coefficients of various polynomials and was wondering if there was a way to reduce the following stuff. Let polynomial, ...
0
votes
0answers
35 views

Prime - number theory [duplicate]

Why is the digit 1 is not a prime number? 1 can be devided by 1 and itself. I think it's because we can express like 1= 1x1x1 ... is it true or not?
2
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0answers
32 views

Notation Question in proof of Kronecker-Weber theorem

I am trying to understand this proof of the Kronecker-Weber theorem by Franz Lemmermeyer http://arxiv.org/pdf/1108.5671.pdf. The set-up is this: $K/\mathbb{Q}$ is a cyclic extension of prime degree ...
2
votes
1answer
50 views

Why is $\mathbb{Q}_\infty = \mathbb{R}$?

Why, in the context of p-adic numbers, do we have the convention $$\mathbb{Q}_\infty = \mathbb{R} \quad$$ ? It must have something to do with the generalization of the Legendre-symbol for ...