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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Show that if $x\equiv 1 \pmod {m^k}, $then $x^m \equiv 1\pmod{m^{k+1}}$.

Let $k\ge 1, m\ge 1.$ Show that if $x\equiv 1 \pmod {m^k}, $then $x^m \equiv 1\pmod{m^{k+1}}$. First I noticed that the assumption would imply $x^m \equiv 1 \pmod{m^k}$, but that doesn't seem to ...
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0answers
21 views

Jacobi Identities

Can anyone guide me how can I prove these two identities? a)$$\prod_{n=1}^{\infty}\frac{1-q^{2n}}{1-q^{2n-1}}=\sum^{\infty}_{n=1}q^{n(n+1)/2}$$ b) ...
1
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1answer
16 views

$a,b,c,p$ are rational number and $p$ is not a perfect cube

Given that $a,b,c,p$ are rational number and $p$ is not a perfect cube, if $a+bp^{1\over 3}+cp^{2\over 3}=0$ then we have to show $a=b=c=0$ I concluded that $a^3+b^3p+c^3p^2=3abcp$ but how can I go ...
1
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1answer
23 views

If $n\equiv 4 \pmod 9$ then $n$ cannot be written as sum of three cubes?

Show that if $n\equiv 4 \pmod 9$ then $n$ cannot be written as sum of three cubes. This might be a silly question but I really don't see it? The thing I ended up was: let $n=a^3 + b^3 + c^3$, ...
4
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1answer
52 views

Zeilberger's potential proof of Fermat's last theorem.

Doron Zeilberger suggested the following potential proof for Fermat's last theorem: Let's define: $$W(n,a,b,c) \equiv (a^n + b^n - c^n)^2$$ I am almost sure that there exists a polynomial, ...
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0answers
16 views

Sums of Two Real Power

I want to find the nontrivial solutions for $m_{1}^{\theta}+n_{1}^{\theta}=m_{2}^{\theta}+n_{2}^{\theta}$ where $m_{1},m_{2},n_{1},n_{2}\in\mathbb{Z}_{\geq0}$ and $\theta\in\mathbb{R}_{>1}$. If ...
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0answers
18 views

Intro to Number Theory, Simple Continuous Fractions Question? [on hold]

I have no idea how to start this question, any help would be appreciated! Show that $k_n|k_{n-1}\alpha-h_{n-1}|+k_{n-1}|k_n\alpha-h_n|=1$
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0answers
15 views

Chevalley-Warning Theorem

In Chevalley-Warning Theorem, are the non-trivial solutions such that all variables' instances are non zero, or at least one variable's instance is non zero? That is: does the counting of solutions ...
4
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2answers
58 views

Explicit Galois Action for $X^3 - X -1$

I have always been frustrated with how indirect discussions of Galois Theory are in Algebra textbooks. Even in fine treatments such as Miles Reid. Are there any good examples where we can draw ...
0
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1answer
43 views

Fermat's last theorem and $\mathbb{Z}[\xi]$

I heard that one can prove special cases of FLT by using unique factorization in $\mathbb{Z}[\xi]$ (whenever this is possible), where $\xi$ is a primitive $n$-th root of unity. How can one do this in ...
3
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4answers
37 views

Elementary theory of numbers and the phi function. [on hold]

Question: Let $n$ be a natural number, and suppose that $2 \phi(n) = n$. Prove that $n$ is a power of $2$.
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2answers
3k views

Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. The following are exluded: Books by mathematical ...
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0answers
25 views

What is the number of points with integer coordinates in in a rectangle? [on hold]

Let $x_1,x_2,y_1,y_2$ be integers and consider a line segment from the point $(x_1,y_1)$ to the point $(x_2,y_2)$. Further, consider the greatest common divisor of $(x_2 - x_1)$ and $(y_2 - y_1)$. ...
2
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1answer
72 views

Gorgeous diophantine equation

How to find all integer solutions of the following equation? $$y^7=14 \cdot 3^{100}x^6 + 70 \cdot 3^{300} x^4 + 42 \cdot 3^{500} x^2 + 2 \cdot 3^{700}$$
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0answers
38 views

the sum of the reciprocals of the primes

The sum of the reciprocals of the primes is $\sum \limits_{p}\frac{1}{p} \approx N \ln\ln(N)$ what about this sum where $p_{3}=3,p_{5}=5,p_{n}=\sum \limits^{N}_{j=5}\frac{1}{p_{j}} \sum ...
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0answers
16 views

What is the significance of $SL(2, \mathbb{R} / SL(2, \mathbb{Z}))$ in studying lattices in geometry of numbers?

I was listening to a talk about lattices and the geometry of numbers and at one point they jumped from discussing a 2d lattice into discussing $SL(2, \mathbb{R})\ /\ SL(2, \mathbb{Z}))$ and it was not ...
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1answer
268 views
+50

(Unsolved) In this infinite sequence, no term is a prime: prove/disprove.

$ 343,~ 34343, ~3434343, ~343434343,\ldots$ $\begin{array}\\ \color{Red}{343} &\color{Red}{: 7^3}\\ 34343 &: 61\times 563\\ \color{green}{3434343} &\color{green}{: 3\times 11^2\times ...
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2answers
149 views
+250

Prove that $\mathcal{O}_3$ and $\mathcal{O}_7$ are euclidean domains

For a non-square integer $d$ such that $d \equiv 1 \mod 4,$ we define the set $$\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}.$$ Prove that ...
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0answers
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Quotient of invariant differentials of an elliptic curve is a constant

I am stuck on page 97 last paragraph of 'Advanced topics in the arithmetic of elliptic curves by Silverman'. I will be glad if someone can explain the following. "The quotient of two non-zero ...
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2answers
22 views

Primitive elements proof

I am looking for the proof of this: Let $p$ be prime. Then there exists a primitive element $g$ modulo $p$. I have done the following: By Fermat's Little theorem, we have that $g^{p-1} \equiv 1 ...
2
votes
2answers
330 views

Proving that any rational number can be represented as the sum of the each cube of three rational numbers

I found the following question in a book: Prove that any integer can be represented as the sum of the each cube of five integers. The answer : ...
0
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1answer
41 views

All integer is a sum of four cubes of rational

I am interested in a classic problem of representation of integers as a sum of four cubes of integers. This problem has been partially solved missing only integers of the form 9k ± 4. I got a proof ...
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1answer
23 views

Is the sequence $(v_p(n))$ of $p$-adic valuations of positive integers the fixed point of a morphism, for every prime $p$?

Fix a prime number $p$ and consider the sequence $\mathbf{v}_p = (v_p(n))_{n \geq 1}$, where $v_p$ is the usual $p$-adic valuation, i.e. $v_p(n) = a$ iff $p^a \parallel n$. While browsing the OEIS I ...
0
votes
1answer
8 views

quotient of lattices, why of finite length? (about a statement in Local Fields of Serre)

I am studying the book "Local Fields" from Serre. At the beginning of chapter III the setting is follows: $A$ denotes a Dedekind domain, $K$ its field of fractions and $V$ is a finite-dimensional ...
10
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1answer
218 views

Is this olympiad-like question about remainders an open problem?

Suppose that we are given two positive integers $x$ and $y$ such that $$x \mod p \leqslant y \mod p$$ for each prime number $p$. (Here, $x \mod p,\; y \mod p$ stand for the least non-negative ...
0
votes
1answer
37 views

Floor inequality with prime

If $a$ and $b$ are positive integers and $a\ge b$ and $b$ is an odd prime, show that: $$\left\lfloor \frac{6a-1}{b}\right\rfloor+\left\lfloor\frac{a}{b}\right\rfloor\ge \left\lfloor ...
0
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0answers
21 views

Prove that there exists an element $a$ in the modulo $p$ multiplication group such that $|a| = p-1$. [duplicate]

Here, $|a|$ is the order of $a \mod p$, i.e., the smallest positive $x$ such that $a^x=1 \mod p$.
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3answers
104 views

Finding integer solutions of $m^2-n^5 = m - n$

How to list all integer solutions of $m^2-n^5 = m - n$ Here $m$ and $n$ are some positive integers. Also, I want to know the name of this type equations (if name exist). Regards Rosy
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0answers
20 views

There is an estimation of a $ \sum_{x\in\mathbb{F}_p} \left(\frac{a_{n}x^n+a_{n-1}x^{n-1}+…a_0}{p} \right) \le (n-1)\sqrt{p}$

Where can I find a proof of the following inequality? ( $n$ is odd) $$ \sum_{x\in\mathbb{F}_p} \left(\frac{a_{n}x^n+a_{n-1}x^{n-1}+...a_0}{p} \right) \le (n-1)\sqrt{\vphantom{d}p} $$ I read that ...
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2answers
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How many numbers smaller that $N$ can be written as a sum of two square numbers?

We define $$a_N =\# \{ n \leq N, \exists (n_1,n_2) \in \mathbb{N}^2, n = n_1^2 + n_2^2 \}.$$ Can we have the exact value of $a_N$, or at least an asymptotic behavior such as $$ \alpha N \leq a_N \leq ...
0
votes
1answer
94 views

Dedekind rings which are UFDs but not PIDs?

I just have a really quick question of an example that I was trying to come up with. Are there any number rings which are UFDs but not PIDs?
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1answer
50 views
+50

What are the bases $\beta$ such that a number with non-periodic expansion can be approximated with infinitely many numbers with periodic expansion?

Warning: This problem requires a bit of setting. Fix a finite set $A \subset \Bbb{Z}$ and consider an infinite non (ultimately) periodic sequence $\mathbf{a}=(a_i)_{i \geq 1}$ of elements of $A$ such ...
0
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1answer
36 views

Continued Fraction

I am working on the following question "Use the continued fraction $[1;0,1,1,2,1,1,4,1,1,6,1,1]$ to get an estimate for $e$." But I got stuck when I tried to compute $q_i$, since $a_1=0$ , $q_1 =0$. ...
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0answers
20 views

Formula for coefficient of Mahonian numbers

I recently came out with this article . It tells about triangle of mahonian number.The T(n,k) is coefficients in expansion of Product_{i=0..n-1} (1 + x + ... + x^i), where k ranges from 0 to n*(n + ...
0
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1answer
32 views

Arithmatic on modular curves

I had tried to read the first few pages of Glenn Stevens's$\,$ Arithmetic on Modular Curves, but it is somehow extremely unreadable to me, the text format is odd and stating too much facts without ...
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1answer
50 views

Pythagorean Triples : Is every positive integer $\gt$ $2$ part of at least one Pythagorean triple?

I was doing some basic number theory problems from Rosen and came across this problem: Show that every positive integer $\gt$ $2$ is part of at least one ...
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2answers
16 views

Fermat numbers of the form of $b^2$

For n > 1 Let $F_n = 2^{2^n} + 1$ be a fermat number and b = $2^{2^{n - 2}}$ * ($2^{2^{n - 1}}$ - 1 ). Then $b^2$ $\equiv$ 2 (mod $F_n$) I tried to square the original expression I got something ...
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0answers
65 views

Sums of three cubes in arithmetic progression equal to a cube $x^3+(x+y)^3+(x+2y)^3 = z^3$

Using exhaustive search, small positive and primitive integer solutions to, $$x^3+(x+y)^3+(x+2y)^3 = 3 x^3 + 9 x^2 y + 15 x y^2 + 9 y^3= z^3\tag1$$ are, $$x,y = 3,1,\quad x+y =2^2$$ $$x,y = ...
22
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2answers
625 views

Odd perfect squares whose decimal representation consist only of 1's and o's

Are there any odd perfect squares (apart from the trivial 1), whose decimal representations only uses 1 and 0? Working modulo 8, we can get that the last 3 digits must be 001. However, since $4251^2 ...
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1answer
16 views

How does simplification work when solving linear combinations?

So I'm currently trying to wrap my head around finding gcd through the Euclidean Algorithm in order to write the integers as a linear combination. For example, a problem is to express the ...
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0answers
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Math: A discovery or a creation? [on hold]

I am just curious as to what math is at it's very basic state. Is math something that humans have invented? Or is it more of a discovery? Or possibly something completely different. If it is something ...
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0answers
50 views

Sublists Conjecture

The conjecture: For those $k$ that have a saturated sublist $a_{j}$, the first occurrence is: $$j \geq k+3.$$ A proof will imply Oppermann and will be a start to a pattern-based attack on the ...
5
votes
1answer
86 views

Pythagorean Triples : Show that exactly one of $x$, $y$, and $z$ is divisible by $5$

I was doing some basic number theory problems from Rosen and came across this problem: Show that if $(x, y,z)$ is a primitive Pythagorean triple, then exactly one of $x$, $y$, and $z$ is divisible ...
3
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1answer
26 views

Factor into primes in Dedekind domains that are not UFD's?

Does it make sense to factor numbers into prime numbers in Dedekind domains that are not unique factorization domains? I can't really see how it would make sense.
3
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1answer
35 views

power sum divisible by prime

$p>2$ is a prime and $p-1$ doesn't divide $n$. Prove that $$1^n+2^n+3^n+\ldots +(p-1)^n \equiv 0 \pmod{p}$$ My solution so far: If $n$ is odd then \begin{align*} 1^n+2^n+3^n+\ldots +(p-1)^n ...
2
votes
5answers
61 views

Possible solutions of a diophantine equation: $p^2+pq+275p+10q=2008$

What are couples of prime integers that verify this diophantine equation: $$p^2+pq+275p+10q=2008?$$ I tried to solve this equation trough the rules of modular-arithmetic. I rewrite the equation as: ...
10
votes
5answers
293 views

Show that $\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n d^{n-1},a-b)$

How to show that $$ \gcd\bigg( {a^n-b^n \over a-b} ,a-b\bigg )=\gcd(n d^{n-1},a-b ) $$ $a,b\in \mathbb Z$ where $d=\gcd(a,b)$? Note $\ $ Some of the answers below were merged from this ...
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vote
0answers
30 views

Integral intersections between quadratic sequences

How can I find the integer solutions to: $$ x^2=\frac{1}{2} n (n+1) $$ By brute force I have found the solutions (6,8) (35,49) and (204,288) but then it gets harder. Note that the perfect squares ...
1
vote
1answer
22 views

Pollard Rho intuition

I have been reading about pollard rho factorization, however their is something I don't understand if we don't use a polynomial that is pick two random numbers and see the gcd(a-b,n) > 1 if it is ...
23
votes
1answer
902 views

Murder at Hilbert's Hotel!

I'm sorry if this is a duplicate in any way. I doubt it's an original question. Due to my ignorance, it's difficult for me to search for appropriate things. Motivation. This question is inspired by ...