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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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 ...
0
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2answers
21 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 ...
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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 ...
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1answer
37 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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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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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
40 views

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 ...
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0answers
19 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 + ...
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1answer
21 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 ...
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1answer
48 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 ...
1
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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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1answer
31 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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2answers
35 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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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
50 views

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 ...
3
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1answer
24 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
33 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 ...
1
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0answers
29 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
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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 ...
2
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4answers
34 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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0answers
30 views

degrees of L-functions and dimensions of Shimura Varieties

I try to grab a (very) little understanding of what a Shimura variety is, and although I still don't understand the formal definition of that notion, a few vague ideas have come to my mind. Hence ...
3
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0answers
17 views

Squarefree products of a class of primes

Numbers which are the sum of two squares are the product of a square and a collection of distinct primes which are 1 or 2 mod 4. Landau proved that there are $\sim kx/\sqrt{\log x}$ such numbers up ...
0
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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 ...
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1answer
39 views

Systems of Congruences Modulo a Prime

Let $p$ be prime and $A,B\subseteq\{{1,\dots,p}\}$ be distinct s.t. $|A|=|B|=\left\lfloor p/2\right\rfloor$ and suppose I have a system of $k$ congruences $$\sum_{i\in A}i^d \equiv \sum _{j\in ...
5
votes
1answer
84 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 ...
0
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0answers
17 views

$k(\mathfrak p)$ basis for $A / pA$

I'm reading this pdf which is showing that a rational prime $p$ ramifies if and only if it divides the discriminant of its number field $K$. I've come across the following line: Let $p \in \mathbb ...
0
votes
1answer
66 views

A class of irrational numbers

Consider the function $f(x) = a^x +b^x – c^x$ with positive coprime integers $a < b < c$. When $f(2) > 0$, we have $f(t) = 0$ for some unique $t > 2$. Using FLT prove that $t$ cannot be ...
2
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3answers
29 views

Make $kt^2+(3k+1)t+4k+1$ constant?

Find $k$ such that $kt^2+(3k+1)t+4k+1=0$ is an identity (i.e. true for all $t$). E.g. $k=t+1$ doesn't work since you end up with a third degree polynomial in $t$ which determines $t$, making $t$ ...
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0answers
32 views

Is this curve an elliptic curve

I want to know whether curve $$x^3+21x^2+35x+7=4xy+4y^3$$ is an elliptic curve or not. If that is an EC so what is its Weierstrass form?
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votes
3answers
100 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
3
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1answer
32 views

A result of Erdos: the multiplicative persistence of $n$ is at most $c\ln(\ln n)$

Multiply all the digits of a number $n$ by each other, repeating with the product until a single digit is obtained. The number of steps required is known as the multiplicative persistence of $n$. ...
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2answers
26 views

Construct a circuit of given equivalent resistance using a set of given resistors?

Here is a fun little problem I thought of. It involves electronic circuits, but it's really a number theory problem. We have a box containing an infinite number of resistors with resistance $R$. We ...
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0answers
29 views

How many prime number sieves are there? [on hold]

How many prime number sieves are there? The sieve of Eratosthenes is well known. Write it as: Eratosthenes: {4,6,8,10,...}+{6,9,12,15,...}+{10,15,20,...}+... But are there other types of ...
2
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1answer
29 views

Modular forms: What is $\mathbb{H} / SL_2(\mathbb{Z})$?

I am beginning to understand the very basics of modular forms, in that I understand the concept of a weakly modular function, I have seen the examples of $G_k(z)$ and $E_k(z)$ as weakly modular ...
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0answers
21 views

Prove that If $p$ is an odd prime, then any divisor of the Mersenne number $M_p = 2^p − 1$ is of the form $2kp + 1$

For example, $M_{11} = (2 · 1 · 11 + 1)(2 · 4 · 11 + 1)$ If $q$ is a prime divisor of $M_p$ then $\exists k \in Z$ such that $qk = 2^p-1$. Now $ord(2,q)$ is the smallest positive integer $a$ such ...
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5answers
54 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 ...
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0answers
1k views

$ \lim_{n \to \infty} \sum_{i=1}^n\{ n / i \} (i/n \{ n / i \} - ln(n) -2){\mu{(i)}} +\sum_{i=1}^n 1/2( log(2 \pi [n/i])){\mu{(i)}} = \beta $

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\{ n / i \} (i/n \{ n / i \} - ln(n) -2){\mu{(i)}} +\sum_{i=1}^n 1/2( ...
0
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1answer
32 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
25 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 ?
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1answer
49 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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2answers
32 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)} ...
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1answer
25 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 $36 \equiv 1 \pmod 7$.
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0answers
50 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 ...
2
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0answers
42 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
29 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 ...
1
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0answers
39 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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votes
0answers
42 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
138 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
63 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
36 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 ...