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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5answers
37 views

Mathematical Induction on a Subset of the Natural Numbers

I am given a strict inequality of the form $$ 2n - 8 < n^2-8n+14, $$ where $n$ belongs to the set of natural numbers $\mathbb{N}$ (in this case $n$ does not equal 0). I am asked, for what values ...
1
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0answers
40 views

Implications of $ \lim_{n \to \infty} \sum_{i=1}^n (\ln(i) -2)\operatorname{frac}(\frac{n}{i})\mu(i) =0 $?

I'm just a physics undergraduate. I think (in an non-rigorous way) I have managed to prove: $$ \lim_{n \to \infty}\sum_{i=1}^n (\ln(i) -2) \operatorname{frac}\left(\frac{n}{i}\right)\mu(i) =0 $$ ...
2
votes
4answers
61 views

Prove $\sum_{i=2}^{n}\frac{1}{(n-1)n}$ = $\frac{(n-1)}{n}$ using induction.

I need to prove $\sum_{i=2}^{n}\frac{1}{(i-1)i}$ = $\frac{(n-1)}{n}$ using induction. I am getting stuck midway through the inductive step. Here is what I have: ...
0
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1answer
23 views

Does there exist an integer $s$ such that every integer $> 1$ can be written as a sum of at most $s$ primes?

Does there exist an integer $s$ such that every integer $> 1$ can be written as a sum of at most $s$ primes ?
3
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1answer
21 views

Is it true that every sufficiently large positive integer can be written as a sum of a square free number and a perfect square ?

Is it true that $\exists k \in \mathbb Z^+$ such that every integer $n >k$ can be written as a sum of a square free number and a perfect square ?
0
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1answer
13 views

Ideals in the ring of gaussian integers of a given norm

What are the ideals in the ring of gaussian integers of a given norm, (say $20$) ? The ring of integers is $\mathbb Z[i]$ and it is a PID, so any ideal must be principal. If the ideal $I$ is ...
2
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1answer
22 views

Giving integer images (bis)

Prove the statement below (the restriction $x < y < z$ is to avoid apparent uncertainties but the property is valid for all $x, y, z$ really). $$F(x,y,z) = \frac{(y+z)x^n}{(z-x)(y-x)} ...
0
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1answer
23 views

Counter example for theorem 2.18 in A computational introduction to number theory and algebra

This is a theorem from V. Shoup. A computational introduction to number theory and algebra. I have a counter-example: $\beta = 6, p = 7$, and $36 \equiv 1 \pmod 7$.
0
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0answers
27 views

Blocks of consecutive natural numbers

Let a < b be natural numbers. Prove that every block of b consecutive natural numbers contains two distinct elements whose product is divisible by ab. Suppose now a < b < c are natural ...
1
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0answers
39 views

How to find whole number answers in systems of square root equations

Given the following 4 equations, can you find 4 whole number answers using whole number variable inputs? $x,y,z$ where $x>y>z$ $Eq 1 = (x^2-2xy+y^2-2xz+z^2)^{\frac{1}{2}} $ $Eq 2 = ...
0
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2answers
27 views

If $x^2 \equiv y^2 \pmod {p^r}$, where $p$ is an odd prime not dividing $x$ or $y$ then $x \equiv \pm y \pmod {p^r}$

If $x^2 \equiv y^2 \pmod {p^r}$ then $p^r \mid x^2 - y^2$ and so $p^r\mid(x-y)(x+y)$. Now since $p$ is a prime that is not dividing $x$ nor $y$ then it's easy to see that $p^r \nmid x$ and also $p^r ...
0
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0answers
33 views

Hardy Littlewood Circle Method

I'm working through Vaughan's book on the Hardy Littlewood circle method, which uses the following lemma: Suppose that $\alpha \geq \beta$ are positive real numbers, and that $\beta \leq 1$. Then: $ ...
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0answers
41 views

Is it true that If $n$ is an integer such that $n > 2$, $x$ and $y$ are positive integer such that $x > y$ , then [on hold]

How to Prove or Disprove: If $n$ is an integer such that $n > 2$, $x$ and $y$ are positive integer such that $x > y$ , then $\sqrt[n]{\frac{x^{2n} - y^2}{4}}$ is not a positive integer.
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votes
4answers
104 views

Can you check my proof of Fermat's Last Theorem? [on hold]

I've come up with a proof of Fermat's Last Theorem and my teacher would not look at it so i was wondering if you could check. I know it's supposed to be hard to prove, but I use a "trick" from calc ...
2
votes
2answers
58 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 ...
1
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0answers
25 views

The Nagell-Ljunngren Equation

I have been trying to find the papers of Nagell and Ljunngren, which deal with the equation $$\frac{x^n - 1}{x - 1} = y^2$$ and solve it completely. Many papers cite these papers, but I haven't found ...
1
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2answers
41 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
votes
1answer
31 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
26 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 ...
2
votes
0answers
21 views

Small proximity of important points of a function

Let $a,b,c$ be coprime integers with c greater than b and a, $a^2 + b^2 \gt c^2$ 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) ...
1
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0answers
17 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
votes
3answers
89 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
votes
2answers
50 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
50 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
votes
1answer
19 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
24 views

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

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
62 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
votes
1answer
22 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 ...
0
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0answers
24 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
vote
1answer
18 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
22 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 ...
-3
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
14 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
votes
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
50 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
113 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
80 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 ...
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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
votes
0answers
27 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?
7
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0answers
74 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
1answer
60 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
votes
1answer
18 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 $o(\bar x^2)=2$ ...
0
votes
0answers
19 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
48 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
votes
1answer
45 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
votes
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
40 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) ...