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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9
votes
2answers
319 views

Making an elliptic curve out of a cubic polynomial made a cube, or $ax^3+bx^2+cx+d = y^3$

What is the transformation such that a general cubic polynomial to be made a cube, $$ax^3+bx^2+cx+d = y^3\tag{1}$$ can be transformed to Weierstrass form, $$x^3+Ax+B = t^2\tag{2}$$ (The special ...
3
votes
0answers
70 views

General question about 'vieta jumping'

Suppose I want to prove that a variable posesses a certain property (e.g. is a square). For example if I wanted to prove that $x$ in $\frac{x^2+y^2+1}{xy} = k$ has the property of being a square (It ...
0
votes
0answers
58 views

What math do I need to find the lowest upper bound for this?

Let $n = pq$ for primes $p,q$. I want to find a lowest upper bound for the positive integer $r$ such that $$\displaystyle\sum_{i=0}^{\lfloor(n-m)/r\rfloor}\binom{n}{q(ri+m)} = 0 $$ modulo $n$. Where ...
1
vote
0answers
105 views

A Gauss sum over a field.

Let $K$ be a field (not necessarily $\mathbb C$) and let $\zeta=\zeta_n$ be a primitive $n$th root of unity in $\bar K$. I would like to know if there is a formula calculating $$ \sum_{k=1}^n ...
16
votes
1answer
339 views

Generalization of $\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$

I have been asking the following question at MSE with an answer: $\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true? I found this relational ...
1
vote
2answers
69 views

Find all $x,y,z \in \mathbb{Q}.x^2+y^2+z^2+x+y+z=1$

Find all $x,y,z \in \mathbb{Q}$ where.$x^2+y^2+z^2+x+y+z=1$
2
votes
1answer
47 views

Proof of $a^{2^p}=x^pax^{-p}=a$ $\forall a \in G$

I have to complete this exercise: "Let $G$ be a group with an element of finite order $n>1$ and exactly two conjugacy classes. Prove that $|G|=2$" The author gives some hints: " Prove the ...
4
votes
1answer
103 views

$1^k+2^k+\cdots+n^k\mod n$ where $n=p^a$.

My friend said that for any $n=p^a$, where $p$ is odd prime, $a$ is positive integer then: If $k$ is divisible by $p-1$ then $1^k+2^k+\cdots+n^k\equiv -p^{a-1}\pmod{p^a}$. I am very sure that his ...
3
votes
2answers
95 views

Extending primes

This question is more of a curiosity than anything. Start with a prime number and consider concatenating digits onto the right hand side. Sometimes you can make a prime and continue the process ...
1
vote
0answers
25 views

(Reference Request) Canonical forms for Real and Complex binary forms of low degree.

I am asking for a reference for Canonical forms for Real (and Complex) binary forms of low degree with respect to the natural action of the Real (and Complex) special linear group $SL_{n}(\mathbb{R})$ ...
1
vote
1answer
398 views

Hausdorff dimension of the set of rational numbers within a certain interval?

Intro: The Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated with any metric space. In general the Hausdorff dimension ...
34
votes
4answers
752 views

$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?

I found the following relational expression by using computer: For any natural number $n$, $$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor.$$ Note ...
11
votes
1answer
381 views

Linear independence of the numbers $\{1,\pi,{\pi}^2\}$

Does someone know a proof that $\{1,\pi,{\pi}^2\}$ is linearly independent over $\mathbb{Q}$ ? The proof should not use that $\pi$ is transcendental. $\{1,e,e^2,e^3\}$ is linearly independent over ...
27
votes
4answers
628 views

Linear independence of the numbers $\{1,e,e^2,e^3\}$

Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent over $\mathbb{Q}$? The proof should not use that $e$ is transcendental. $e:$ Euler's number. $\{1,e,e^2\}$ is linearly ...
7
votes
1answer
340 views

Ring class field of $\mathbb{Q}(\sqrt{-19})$

I am looking an explicit form for the ring class field of the order $\mathbb{Z}[\sqrt{-19}]$ in the quadratic field $\mathbb{Q}(\sqrt{-19})$. Does anyone know if there is some and how it is?
14
votes
2answers
228 views

Solve $(a^2-1)(b^2-1)=\frac{1}4 ,a,b\in \mathbb Q$

Does the equation $(a^2-1)(b^2-1)=\dfrac{1}4$ have solutions $a,b\in \mathbb Q$? I search $0<p<1000,0<q<1000$, where $a=\dfrac{p}q$, but no solutions exist. I wonder is this equation ...
9
votes
3answers
294 views

The integer $c_n$ in $(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$

For non-negative integer $n$, write $$(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$$ where $a_n,b_n,c_n$ are integers. For any non-negative integer $m$, prove or disprove ...
4
votes
1answer
138 views

Does $a^6+b^6 = c^6+d^3$ have a non-trivial solution?

It is conjectured that, $$x_1^8+x_2^8+x_3^8 = y_1^8+y_2^8+y_3^8\tag{1}$$ has no non-trivial solutions. However, if we relax it a bit then, $$x_1^8+x_2^8+x_3^4 = y_1^8+y_2^8+y_3^4\tag{2}$$ can be ...
3
votes
2answers
116 views

Modulo Problem, Fermat's little theorem

Find the value of the unique integer x satisfying $O \le x \le 17$ for which $$ 4^{1024000000002} \equiv x\pmod{17} $$ I think this is related to Fermat's little theorem. I'm knowledgeable with the ...
1
vote
1answer
99 views

A problem about divisibility: Partition a number into two and three digits

Now I have proved the following two problems: (1) Prove that a number and the sum of its digits have the same remainder upon division by 9. (2) Given an interger, consider the difference ...
6
votes
1answer
199 views

Prime harmonic series

We have following identity: ($p$ is a prime number) $$\left(1+\frac{1}{p}\right)\sum_{k=0}^n\frac{1}{p^{2k}}=\sum_{k=0}^{2n+1}\frac{1}{p^k}$$ Now, How to derive the following inequality from the above ...
3
votes
1answer
94 views

Expressing integers as a sum of squares

There have been many results about the number of squares needed to represent a positive integer. Lagrange's four-square theorem tells us that $4$ squares suffice for any integer and there have been ...
2
votes
0answers
49 views

Abelian SubGroup Variant:

Consider the following problem: Find integers $x_1, x_2, x_3,\dots, x_n$ Such that: $$P(x_1,x_2,\dots, x_n) = Q$$ for some integer $Q$ and polynomial $P$ where for all permutations of any set of ...
8
votes
1answer
190 views

Quadratic extensions of $\mathbb Q$

This is a question from Lang's ANT, Thm 6, ch.IV, $\S2$. It states that every quadratic extension of $\mathbb{Q}$ is contained in a cyclotomic extension and that it's a direct consequence of the ...
3
votes
1answer
47 views

If $x^2+ax+b$ is an integer for every integer $x$ then comment on the coefficients $a$ and $b$ MCQ

Probably a more general category (number theory) multiple choice question but no clue how to get to a clear conclusion . Here's how it goes : Q) If $x^2+ax+b$ is an integer for every integer x ...
3
votes
3answers
755 views

How many factors does 6N have?

Given a number $2N$ having 28 factors another number $3N$ having 30 factors, then find out the number of factors of $6N$.
5
votes
3answers
1k views

Proof for Euler's Beta function for positive integers

I googled this, but I didn't really find anything good. So I just want a proof that for positive integers $x$ and $y$: $$\int_{0}^{1} t^{x-1} \cdot (1-t)^{y-1} dt = \frac{(x-1)! \cdot ...
5
votes
1answer
117 views

Solutions of a cubic diophantine equation in $\mathbb{Z}/p\mathbb{Z}$

Suppose $p\in\mathbb{Z}$ is prime and $p\equiv 1\pmod{3}$. Is there an estimate of the number of solutions of $x^3+y^3=z^3$ in $\mathbb{Z}/p\mathbb{Z}$, preferably using elementary number theory and ...
0
votes
1answer
88 views

Divisors of $q^kp^r$

This is a generalization of my previous problem. Let $p$ and $q$ be prime numbers. What is the necessary and sufficient condition (in terms of $p,q$ and $k,r$) such that we can partition the divisors ...
6
votes
3answers
143 views

Determine the minimum of $a^2 + b^2$ if $a,b\in\mathbb{R}$ are such that $x^4 + ax^3 + bx^2 + ax + 1 = 0$ has at least one real solution

I just wanted the solution, a hint or a start to the following question. Determine the minimum of $a^2 + b^2$ if $a$ and $b$ are real numbers for which the equation $$x^4 + ax^3 + bx^2 + ax + 1 = 0$$ ...
1
vote
2answers
76 views

Integer outputs of $y=x^2$ , do their last digits form an irrational?

Let the domain of $y=x^2$ be the positive integers. I input consecutive positive integers from $[1, \infty)$ their last digits are $a, b, c, ...$ respectively. If I then make the number $z=\frac ...
6
votes
1answer
136 views

What is the smallest real $q$ such that there is always a prime between $n^q$ and $(n+1)^q?$

In this answer, it is mentioned that for $q=3$, we are guaranteed the existence of a prime between $n^q$ and $(n+1)^q$, and that it is conjectured that this is true for $q=2$. I am wondering though, ...
2
votes
1answer
62 views

If $p$ is a prime number, prove that for any $a \in \mathbb{Z}$, we have $p |a^p+(p-1)!a$ and $p|(p-1)!a^p+a$

If $p$ is a prime number, prove that for any $a \in \mathbb{Z}$, we have $$p |a^p+(p-1)!a$$ and $$p|(p-1)!a^p+a$$ I totally got no idea how to start. Can anyone give some hints?
3
votes
1answer
111 views

Krull's intersection theorem in the q-expansion principle

I'm currently reading the proof of the q-expansion principle in Katz'73 paper "p-adic properties of modular schemes and modular forms" . The principle itself is a Corollary (1.6.2) of Theorem 1.6.1, ...
3
votes
1answer
115 views

Is there any kind of irrational number wich does not contain digit 9?

At first we must prove that there is or is`t irrational numbers which does not contain digit 9! if there are many kind of such numbers, then there is another question: how to write down algebraic ...
0
votes
3answers
335 views

how to find out any digit of any irrational number?

We know that irrational number has not periodic digits of finite number as rational number. All this means that we can find out which digit exist in any position of rational number. But what about ...
4
votes
2answers
638 views

How prove this $[\sqrt{23n}]\{\sqrt{23n}\}>3$

show that for any positive integer $n\ne 23m^2,m\in N$, have $$[\sqrt{23n}]\{\sqrt{23n}\}>3$$ and $\{x\}=x-[x]$ I have post this How prove this $|\{n\sqrt{3}\}-\{n\sqrt{2}\}|>\frac{1}{20n^3}$ ...
2
votes
1answer
44 views

Differentiating between prime/semi-prime and other integers

Does there exist a test that checks if a number is prime or a semi prime in polynomial time? I am aware that AKS can be used to check primality but what about semi primality? ...
12
votes
1answer
168 views

What is the significance of the power of $3$ in the sequence of primes given by $\lfloor A^{3^n}\rfloor ?$

Mill's constant is a number such that $\lfloor A^{3^n}\rfloor$ is prime for all $n$. The existence of such an $A$ was proven in $1947$. I know little about number theory, but I am curious as to why ...
1
vote
4answers
415 views

Rational solutions for $x^2+y^2=3$

Are there any rational numbers $x$ and $y$ such that $x^2+y^2=3$. I think there are no rational solutions, but I haven't been able to prove it.
1
vote
0answers
68 views

a diophantine equation from Stewart and Tall [duplicate]

This is from Stewart and Tall from the chapter on Kummer's Theorem. Show that there are no non trivial (non-zero) solutions to $x^3 + y^3=3z^3$
3
votes
1answer
542 views

Characterization of integers which has a $2$-adic square root

Does anyone know an "elementary" proof of the following theorem? Let $k \neq 0$ be a rational integer. Then $k$ admits a square root in $\mathbb{Z}_2$ if $k = 4^a (8b+1)$ for some $a \in \mathbb{N}$, ...
2
votes
0answers
48 views

Number of primes of type 4*n +1 in a range

I want to find number of primes which are congruent 1 (mod 4) in a range [a, b]. The range can be of order $10^9$ as a and b can be from $1$ to $10^9$. I tried segmented sieve but for a range so ...
8
votes
2answers
181 views

Solve $x^2+y^2=2$ for $x,y\in\mathbb Q$.

Solve $x^2+y^2=2$ for $x,y\in\mathbb Q$. I think the answer should be in terms of 1 integer variable $\in\mathbb Z$ only. I rewrite the equation to $(x+y)^2+(x-y)^2=2^2$, then by the formula of ...
1
vote
0answers
72 views

For squarefree $i$ what is $\sum_1^{n} \frac{1}{i}$?

For squarefree $i$ what is $\sum_1^{n} \frac{1}{i}$ ? I use $\sum_{\sqrt{n}>m>1} \mu(m) ln(\frac{n}{m^2}+\frac{1}{2})$. I know about the connection with $\zeta(2)$ and ...
2
votes
0answers
164 views

Approximation to $\pi(x)$ conjecture.

A friend conjectured that $\left[\prod_{k=1}^{a_j <\sqrt{x}} \left(1-\frac{1}{a_k}\right)\right] x$ is usually closer to $\pi(x)$ than $\operatorname{Li}(x)$ is for some (fixed) sequence of ...
4
votes
1answer
777 views

The Diophantine equation $x^2 + 2 = y^3$

How to solve the Diophantine equation $x^2 + 2 = y^3$ with $x,y>0$ ? ($x,y$ are integers.)
2
votes
1answer
91 views

This correct this demonstration of Number theory (binomial Expressions)

$$\\$$Em minha apostila tem as demonstrações dos seguintes lemas:$$\text{Lema (*): Sejam $a,m,n,q,r\in\mathbb{N}$ com $a\geq2$ tais que $m=nq+r$ then:}\\(a^m-1,a^n+1)=\begin{cases}(a^n+1,a^r-1)& ...
0
votes
2answers
69 views

Number obtained by reversing digits. Find the value satisfying a condition.

Let $a$ and $b$ be two-digit integers such that $b$ is obtained by reversing the digits of $a$. The integers $a$ and $b$ satisfy $a^2-b^2=m^2$ for some positive integer $m$. Which could be value of ...
1
vote
4answers
71 views

Determining the general form of $10^x \bmod 210$

While solving a problem I came across solving $10^x\bmod 210$ for various values of $x$. It seems that the values repeat after an interval of 6 for $x\geq4$. Can any one explain how can solve this ...