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.

learn more… | top users | synonyms (1)

14
votes
7answers
916 views

Find all even numbers that can be represented as a difference of squares in only two ways

I am currently working on this proof. I am looking to find (with proof) all even numbers that can be represented as a difference of squares in only two ways. My thoughts thus far. I examined the ...
2
votes
2answers
35 views

Embeddings of pure cubic field in complex field

I know that the complex embeddings (purely real included) for quadratic field $\mathbb{Q}[\sqrt{m}]$ where $m$ is square free integer, are $a+b\sqrt{m} \mapsto a+b\sqrt{m}$ $a+b\sqrt{m} \mapsto a-b\...
0
votes
2answers
67 views

Positive integers satisfying $\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}=\frac{1}{\sqrt{20}}$

Find all the positive integers satisfying $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}=\frac{1}{\sqrt{20}}$$ How to approach this question? I am not getting how to initiate the problem.
0
votes
0answers
36 views

Growth of $\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$ with $r_{0}(n)=\inf\{r\ge 0, (n-r,n+r)\in\mathbb{P}^{2}\}$

Assuming Goldbach's conjecture, let's denote by $r_{0}(n):=\inf\{r≥0,(n−r,n+r)\in\mathbb{P}^{2}\}$ and by $k_{0}(n)$ the quantity $\pi(n+r_{0}(n))−\pi(n−r_{0}(n))$. It seems (see ideasfornumbertheory....
2
votes
1answer
53 views

How to factor numbers that are the product of two primes

What are techniques to factor numbers that are the product of two prime numbers? For example, how would we factor $262417$ to get $397\cdot 661$? Would we have to guess that factorization or is there ...
0
votes
0answers
32 views

How can I solve this recurrence relation for generating triangle-squares?

$$N_k = 17N_{k-1} + 6(8N^2_{k-1} + N_{k-1})^{1/2} + 1$$ $$k\geqslant 1$$ I'm trying to convert a recurrence relationship for producing triangle square-numbers into a closed-form expression in terms of ...
1
vote
1answer
39 views

Condition on $(a_n)_{n \in \mathbf{N}}$ for Dirichlet-series $\sum_{n = 1}^\infty \frac{a_n}{n^s}$ to converge for $Re(s) > 1$.

I'm curious about the conditions on a sequence $(a_n)_{n \in \mathbf{N}}$ of real numbers such that the Dirichlet-series $\sum_{n = 1}^\infty \frac{a_n}{n^s}$ converges absolutely for $Re(s) > 1$. ...
1
vote
0answers
59 views

Is this stronger Kintchine theorem true?

Let $\phi(n)$ be an increasing real valued function on the positive integers. Suppose that almost every $x \in (0,1)$ has $a_n \geq \phi(n)$ for infinitely many $n$, where $a_n$ is the n'th integer ...
3
votes
1answer
34 views

Chowla's Construction of prime having least quadratic non-residue $\gg \log p$

This paper by NC Ankeny mentions that " S. Chowla has proved that there exist infinitely many primes $k$ where the first $c_1\log k$ residues $(\bmod k)$ are all quadratic residues". I recently ...
1
vote
2answers
82 views

Find the probability that $1984!$ is divisible by $n$

Let $a,b,c,d$ be a permutation of the numbers $1,9,8,4$ and let $n = (10a+b)^{10c+d}$. Find the probability that $1984!$ is divisible by $n$. I was told this could be solved by casework on $a$ and ...
4
votes
1answer
69 views

Can $x^2+y^2,y^2+z^2,z^2+x^2$ and $x^2+y^2+z^2$ all be square numbers?

I know that if we want $x^2+y^2$ to be square number, we are looking for pythagorean triple; if we want $x^2+y^2+z^2$ to be a square number, we are looking for pythagorean quadruple. But have we ever ...
4
votes
2answers
76 views

Proving that $\lceil f(x) \rceil$ $=$ $\lceil f(\lceil x \rceil )\rceil$ when $f(x) =$ integer $\implies x =$ integer

On P. 71 in 'Concrete Mathematics' the following Theorem is given: Let $f$ be any continuous, monotonically increasing function on an interval of the real numbers, with the property that \begin{...
0
votes
0answers
46 views

Is a tight concrete bound for the error-term in the prime-number-theorem known?

Here : https://en.wikipedia.org/wiki/Prime_number_theorem it is mentioned that $$\pi(x)=Li(x)+O(xe^{-a\sqrt{ln(x)}})$$ What is a tight upper bound for $|\pi(x)-Li(x)|$ in concrete terms ? The ...
3
votes
1answer
83 views

Solving $a^b = b^a$ for $a,b \in \Bbb N$ where $a,b$ are distinct

While preparing for the Putnam math competition my teacher listed the following problem: Solving $a^b = b^a$ for $a,b \in \Bbb N$ where $a,b$ are distinct. I suspect the answer is that there is only ...
2
votes
2answers
105 views

Is it possible that $a^2+b^2+c^2 = d^2+e^2+f^2$?

Let $a,b,c$ be nonnegative integers such that $a \leq b \leq c, 2b \neq a+c$ and $\frac{a+b+c}{3}$ is an integer. Is it possible to find three nonnegative integers $d,e,$ and $f$ such that $d \leq e \...
0
votes
1answer
24 views

Need help with the proof of a theorem about Gaussian integers

Theorem 6-3. If $\alpha$ and $\beta$ are integers of $Z[i]$, and $\beta \neq 0$ then there are $\kappa$ and $\rho$ in $Z[i]$ such that $$\alpha =\beta\kappa+\rho, \text{ } N_\rho < N_\beta$$ ...
1
vote
2answers
70 views

Notation for representing ANY number?

i'm working on a mathematics/number-manipulation program, and i was wondering if you could practically have a representation that could holds the value of any number. This would need to include ...
0
votes
0answers
26 views

Deriving the sequence if the generating function is irreducible?

I am trying to better understand generating functions and how they can be derived / manipulated / etc. Right now I am operating on this identity, slightly modified from the answer here: For a ...
5
votes
3answers
127 views

Solving a system of linear congruences

Find all positive integer solutions to \begin{align*}x &\equiv -1 \pmod{n} \\ x&\equiv 1 \pmod{n-1}. \end{align*} I rewrote the system as $x = nk_1-1$ and $x = (n-1)k_2+1$. Thus, we have $...
0
votes
1answer
27 views

Integer solution to an hyperbola equation

Given the general equation of an hyperbola $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $ where $B^2-4AC>0$ is it possible to find all integers solutions $(x,y)$ as a function of $A, B, C, D$ and $ F $ ?...
4
votes
2answers
34 views

Smallest chain of consecutive integers not all coprime

Let $t$ be a positive integer. What is the smallest $t$ for which we can find an integer $a$ such that each element of the set $\{a+1,a+2,\dots ,a+t\}$ is not coprime with all other elements of the ...
3
votes
2answers
112 views

characterisation of $n$ as prime using min values of $x$ such that $nx+1$ or $nx$ is square

Let $n\ge 5$ be an odd integer and $k\ =\ \min\{x\in\mathbb{N}\colon nx+1\text{ is a perfect square}\}$ $l\ =\ \min\{x\in\mathbb{N}\colon nx\text{ is a perfect square}\}$ Prove that $n$ is a prime ...
1
vote
0answers
19 views

Does Dirichlet's theorem on arithmetic progressions work with other notions of density?

Dirichlet's theorem on arithmetic progressions is that the density of the prime numbers for each residue class $a$ mod $n$ with $(a, n) = 1$ is $\varphi(n)^{-1}$. I believe this is easier to prove ...
2
votes
1answer
57 views

How do I get a sequence from a generating function?

For example if I have the generating function $\frac{1}{1-2x}$ then it corresponds to the sequence $1 + 2x + 4x^2 + 8x^3 +~...$. I know how to start from the sequence and get the generating function, ...
0
votes
1answer
26 views

Sequence of natural numbers such that $a_i = \sum_{r=1}^{i+4} d_r$

Find a sequence of natural numbers $a_i$ such that $\displaystyle a_i = \sum_{r=1}^{i+4} d_r$ where $d_r \neq d_s$ for $r \neq s$ and $d_r$ divides $a_i$ for all $r \in \{ 1, 2, \dots, i+4\}.$ ...
1
vote
0answers
48 views

a consequence of Prime Number Theorem

By Prime Number Theorem we have $\lim_{n\to\infty}\frac{p_{n+1}}{p_n}=1$, so $\frac{p_{n+1}}{p_n}=1+a_n$ where $a_n\to 0$. How fast does $(a_n)$ converge to $0$ ? Does for example $a_n\ln n$ or $...
2
votes
0answers
51 views

lower bound for sum of distinct n-th roots of unity

Given a positive integer $n$, define $\zeta = e^{2\pi i/n}$ and define $s: \mathbb Z^n \to \mathbb C$$$s(\vec x) = \sum_{k=0}^{n-1} x_k \zeta^k$$ Let us consider the set $S = \{ |s(\vec x)| : \vec x \...
2
votes
3answers
44 views

Find all the integer pairs $(r,s)$ that satisfy $s= (r^2 +3r +8) / (r^2 +r -2)$?

I have been trying to solve this question but struggling to see where to start. Examples I've seen that works are the pairs: $(-3,2) , (4,2), (0,-4)$
1
vote
2answers
54 views

Multiple of 3 as Sum of 4 Cubes

Prove that every multiple of 3 can be expressed as sum of four integer cubes. When working with numbers of form $6n$, a very clear pattern emerged, which the equation below proves elegantly - $\...
3
votes
1answer
62 views

Longest sequence of primes where each term is obtained by appending a new digit to the previous term

What is the longest known sequence of primes where each new term is obtained by appending a new decimal digit to the previous term? Examples: $$(2,23,233,2333,23333)$$ There are no more members in ...
16
votes
1answer
235 views

Why are $L$-functions a big deal?

I've been studying modular forms this semester and we did a lot of calculations of $L$-functions, e.g. $L$-functions of Dirichlet-characters and $L$-functions of cusp-forms. But I somehow don't see, ...
3
votes
1answer
28 views

Returning a decimal number between 0 to 1 for showing how large each number in a set of number is

I hope the title does not go so far, I just want to describe what I want simply: I have a set of random numbers, and I want to return a decimal number from 0 to 1 to show how big (max) the number is. ...
1
vote
1answer
28 views

What is the definition for totally ramified extension for a global field?

What is the definition for totally ramified extension for a global field? For local fields it means the maximal prime ideal generated from the uniformizer totally ramifies. But what is the definition ...
1
vote
1answer
48 views

Prove $\sum_{k=1}^{n} 2^n \text{ mod }k > 2n$ where $n > 1000$

This problem is taken from a Russian textbook of past Olympiads. Its statement looks like this : Given a natural number $n > 1000$ prove that $\sum_{k=1}^{n} 2^n \text{ mod }k > 2n$. ...
0
votes
1answer
45 views

How do you prove that any two integers will always have a greatest common divisor?

I think my book actually prove it by showing that the set of all linear combinations of two integers is a principal ideal of integers. But I can't see how this proves that two integers will always ...
4
votes
1answer
60 views

Subsets with numbers such that $a+b = c$

The set $\{1,2,\ldots,49\}$ is divided into three subsets. Prove that at least one of these subsets contains three different numbers $a,b,c$ such that $a+b = c$. Assume not, so that $\{1,2,\ldots,49\...
0
votes
0answers
17 views

Multivaritate version of Fermat's little theorem

If $f$ is irreducible over $\mathbb{F}_p[x]$ of degree $d$, then $$g(x)^{p^d} \equiv g(x) \bmod f(x)$$ and $p^d - 1$ is the order of the cyclic group $\left( \mathbb{F}_p[x]/f(x) \right)^{\times}$. ...
0
votes
0answers
30 views

Can the 290 Theorem be refined/sharpened to include special conditions?

The 290 theorem states If a positive-definite quadratic form with integer coefficients represents the twenty-nine integers $1$, $2$, $3$, $5$, $6$, $7$, $10$, $13$, $14$, $15$, $17$, $19$, $21$, ...
4
votes
1answer
51 views

Volume in higher dimensions

Let me first state the statement which I want to prove (encountered while studying "Geometry of Number"): Suppose $A$ is a convex, measurable, compact and centrally symmetric subset of $\...
3
votes
0answers
65 views

Prove that the recursion is an integer

A "number triangle" $(t_{n, k})$ $(0 \le k \le n)$ is defined by $t_{n,0} = t_{n,n} = 1$ $(n \ge 0),$ $$t_{n+1,m} =(2 -\sqrt{3})^mt_{n,m} +(2 +\sqrt{3})^{n-m+1}t_{n,m-1} \quad (1 \le m \le n).$$ ...
1
vote
2answers
38 views

A question concerning the Euclidean algorithm

Given a pair of relatively prime integers $m$ and $n$, with $|m| + |n| > 1$, can I always find integers $a$ and $b$ such that am + bn = $\pm$ 1 and $|m| + |n| > |a| + |b|$.
1
vote
2answers
41 views

The product of five consecutive positive integers cannot be the square of an integer

Prove that the product of five consecutive positive integers cannot be the square of an integer. I don't understand the book's argument below for why $24r-1$ and $24r+5$ can't be one of the five ...
-2
votes
4answers
53 views

Set of five consecutive integers

Prove that in any set of five consecutive integers there exists a number not divisible by $2$ or $3$. I thought of doing a proof by contradiction. That is, suppose that each number is either a ...
1
vote
4answers
34 views

modular arithmetic help [closed]

$3t_1 \equiv 1 \pmod 5$ $t_1 \equiv 2 \pmod 5$ how can we derive line 2 from line 1?
3
votes
0answers
27 views

Add $P$ to itself $N$ times on elliptic curve $y^2 = f(x)$, end up with expression in denominator of $x$ vanishing iff $NP$ is point at infinity?

See the second to last paragraph from page 39 of Koblitz's Introduction to Elliptic Curves and Modular Forms. Why is it that when we add a point $P$ to itself $N$ times on an elliptic curve $y^2 = ...
0
votes
1answer
25 views

Decimal expansion of an irrational number not ending in a particular sequence

Consider the set $N$ of natural number, $N$={1,2,3,4,5,6,7,8,9,10,11...} and consider the subsequence {$N_i$} ,$i \in N$ Each $N_i$ consists of elements in ascending sequence greater than $i$ $N_1$...
0
votes
1answer
74 views

Can you Prove or Disprove this?

In $a^n+b^n=c^n$ , $(a<b<c)$ , $a,b,n$ belongs to natural numbers, If $n>=b/2$ , $c$ lies between $(b,b+1)$. Also, only for $n=1,2$ , $c=b+1$.
3
votes
0answers
88 views

$\sqrt[n]{m}$ is irrational if $m$ is not the nth power of an integer

I'm reading the book A Classical Introduction to Modern Number Theory by Ireland and Rosen, and I think that there is an exercise which is false. The exercise says: Prove that " $\sqrt[n]{m}$ is ...
2
votes
1answer
39 views

$a \in \mathbb{Z}[i]$ is a unit if and only if $a$ divides every element of $\mathbb{Z}[i]$? [closed]

As the question title suggests, how do I see that $a \in \mathbb{Z}[i]$ is a unit if and only if $a$ divides every element of $\mathbb{Z}[i]$?
3
votes
1answer
26 views

Does it necessarily follow that $a = ub$ for some unit $u \in \{\pm1, \pm i\}$? [closed]

Suppose that $a$, $b \in \mathbb{Z}[i]$ satisfy $a \mid b$ and $b \mid a$. Does it necessarily follow that $a = ub$ for some unit $u \in \{\pm1, \pm i\}$?