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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0
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2answers
31 views

What delimits the mathematical framework within which information compression limits (from entropy) are valid.

Lets suppose for absurd that I eliminate one number from the naturals. If I were supersticious I would eliminate number 13. Now imagine that to keep normal mathematics possible within such system ...
1
vote
1answer
14 views

Is this Bertrand's postulate-related statement valid?

Bertrand's postulate says: For every $n>1$ there is always at least one prime $p$ such that $n<p<2n$. Is the following statement: For every $n>3$ there is always at least one ...
4
votes
2answers
58 views

Integers of the form $x^2+2y^2$.

I'm stuck in the following problem: prove an integer $n$ is of the form $x^2+2y^2$ if and only if every prime divisor $p$ of $n$ that is congruent to $5$ or $7\bmod8$ appears with an even exponent. I ...
0
votes
1answer
23 views

All possible variants of representation natural number N as product of natural numbers

Task : describe a predicate (on Prolog) that count all possible variants of representation of natural number N as product of natural numbers. For example, 6 = 6*1 = 2*3, so answer is 2. The program ...
2
votes
1answer
24 views

Lower bounds on the index of $\mathbf Z[X]/(P)$ in the ring of integers of a number field

Let $P$ in $\mathbf Z[X]$ be an irreducible polynomial. Let $\mathcal O$ be the ring of integers of the number field $K:=\mathbf Q[X]/(P)$ and $i$ be the index of $\mathbf Z[X]/(P)$ in $\mathcal O$. ...
2
votes
1answer
32 views

Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$

I'm currently reading Andreescu and Andrica's Number Theory: Structures, examples and problems. Problem 1.1.7 states : Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$. The ...
1
vote
0answers
16 views

Why is $K^{\ast n}$ contained in the norm group?

http://www.bprim.org/cyclotomicfieldbook/rlmain.pdf In section 5, $K$ is a local $p$-adic field containing the $n$th roots of unity, and $L = K(\sqrt[n]{x} : x \in K^{\ast})$. Kummer theory tells us ...
2
votes
3answers
55 views

How do I find the factorial of a decimal [duplicate]

How do I find the following: $$(0.5)!(-0.5)!$$ Can someone help me step by step here?
-3
votes
3answers
162 views

Are the irrationals + zero an additive group? [duplicate]

Is it true that the only way two irrationals can sum to a rational is if they sum to zero? Thanks!
5
votes
1answer
37 views
1
vote
0answers
11 views

Special $\omega(n)$-sequence

Let $k$ be a natural number, $\omega(n)$ the number of distinct prime factors of $n$. The object is to find a number $n$ with $\omega(n+j)=j+1$ for each $j$ with $0\le j\le k-1$. In other words, a ...
1
vote
1answer
18 views

Quadratic Diophantine equations on the ring of polynomials

The set of solutions of quadratic equation $a^2+b^2=c^2$ on $\mathbb{Z}$ can be described by Pythagorean triples up to multiplication. Can I use similar results on the ring of integer coefficient ...
2
votes
1answer
36 views

for any positive integer $a,b,n$,and $(a,b)=1$,Is $\frac{1}{a+b}+\frac{1}{a+2b}+\cdots+\frac{1}{a+nb}$ non integer,and How to prove that?

It's easy to prove that both $\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$ and $\frac{1}{3}+\frac{1}{5}+\cdots+\frac{1}{2n+1}$ are nonintegers by multiply $2^k$and $3^k$, and how about the ...
-1
votes
0answers
36 views

If $m^n -m=(m-n)!$, where m>n>1 and $ m=n^2$, then the value of $m^2 +n^2 =?$

If $m^n -m=(m-n)!$, where m>n>1 and $ m=n^2$, then $m^2 +n^2 =?$ how can I find this one ?? I found ridiculous after some stapes..
5
votes
1answer
76 views

theorem relating mersenne numbers?

For $(x2^9)^2=2^q-1+y^2q^2$,where $q$ is prime, is it possible to show that there exists only an unique solution for the pair $\{x,y\}$?
-4
votes
2answers
88 views
1
vote
1answer
27 views

What is known about multi-perfect numbers?

It is unknown if odd perfect numbers exist and it is known that the even perfect numbers are those of the form $$2^{n-1}(2^n-1)$$, where $2^n-1$ is a (Mersenne-)prime. But what is known about the ...
0
votes
6answers
34 views

$\gcd(a,n)\neq 1 \implies $ there is $b$ such that $ab\equiv 0 \pmod{n}$

I have that $\gcd(a,n)\neq 1$ ($a$ and $n$ are not coprime). Then, somehow, I need to prove that exists an $b$ such that $$ab\equiv 0 \pmod{n}$$ What I did: $$ab\equiv 0 \pmod{n}$$ is the same ...
5
votes
1answer
57 views

Extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, splitting.

In the extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, must principal prime ideals of $\mathbb{Z}[\sqrt{-5}]$ necessarily split into 2? Must nonprincipal prime ideals not split? ...
1
vote
1answer
54 views

How to prove that the ring of algebraic integers is a Bézout domain?

I was told that the ring of all algebraic integers (that is, the complex numbers which are roots of a monic polynomials with integral coefficients) is a Bézout domain. But I have no idea how to prove ...
4
votes
0answers
32 views
+50

The asymptotic upper density of $\{xy \colon 1\le x\le y\le 2x\}$

Find the asymptotic upper density of the set $\{xy\, \colon\, 1\le x\le y\le 2x\}$. In other words, let $S$ be the set of all integers which can be expressed as $xy$, for some positive integers ...
5
votes
1answer
105 views

Prove by combinatorial method that $ \frac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $ is an integer [duplicate]

Prove that $$ \dfrac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $$ is a positive integer, where $(m,n) \in \mathbb{Z^{+}}$ I have already solved it using Legendre's Formula ...
2
votes
0answers
27 views

Convergence of series involving Euler's totient function.

I have to show that if $\phi$ is Euler's totient function, then the series $\sum\limits_{n=2}^{\infty} \frac{1}{\phi (n) \log n}$ diverges and $\sum\limits_{n=2}^{\infty} \frac{1}{\phi(n) \log^2 n}$ ...
3
votes
0answers
29 views

An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
2
votes
2answers
50 views

How to show that $(2, \sqrt{82})$ in $\mathbb{Z}[\sqrt{82}]$ is not pricipal?

I tried the obvious things, like using the norm and trying to show that there were no integer solutions to $a^2 - 82b^2 = 2$, but didn't get anywhere. (A friend asked me this.)
0
votes
0answers
17 views

On (known) applications of fixed point theorems to some conjectures in elementary number theory

Let $\sigma$ be the classical sum-of-divisors function. Call an integer $n$ almost perfect if $\sigma(n)=2n-1$. The only known examples are $n=2^k$ for $k \geq 0$. Let $I(n)=\sigma(n)/n$ be the ...
3
votes
1answer
58 views

Why are there so many conjectures in number theory and comparatively less in others?

My question is that : Why are there so many conjectures in elementary number theory and comparatively less in others? This is particularly weird because every topic in maths should have its equal ...
0
votes
1answer
58 views

Why is $\frac{4k -1}{2} \equiv 1 \pmod 4$?

Why is $\frac{4k -1}{2} \equiv 1 \pmod 4$? I need some help with understanding this... The original problem was: find the set of Numbers in which $ f(z)=\frac{1}{1- sin(z)} $ is not defined. This ...
25
votes
1answer
639 views

On Shanks' quartic approximation $\pi \approx \frac{6}{\sqrt{3502}}\ln(2u)$

In Mathworld's "Pi Approximations", (line 58), Weisstein mentions one by the mathematician Daniel Shanks that differs by a mere $10^{-82}$, $$\pi \approx ...
1
vote
1answer
64 views

Number theory / decimal representation

Prove that for any $n\in\mathbb{N}$ there exists a number $m\in\mathbb{N}$ such that the decimal representation of $m^2$ has $n$ ones at the beginning and some combination of $n$ ones and twos at ...
6
votes
0answers
77 views
+50

Does there exists a positive $t$ that satisfy this given condition?

I am curious about the validity of my claim concerning the equations: $(2k-1)t+1$ (1) $(2k^2-2k)t+(2k-1)$ (2) where $k=2,3,4,...$ My claim is for almost all $k$ or for infinitely many $k$, there ...
0
votes
0answers
17 views

On Diophantine approximation and irrationality proofs

This question is an offshoot from this previous MSE post. I have a ratio of two numbers $a$ and $b$ (presumably both positive integers), where $a$ and $b$ are determined by some arithmetic / ...
1
vote
2answers
67 views

Finding de Dr. Bronowski's number

This is a puzzle, created by Dr. J. Bronowski. All credit for this problem goes to him. Find the least positive integer such that moving the leading digit to the end produces a new integer that is ...
5
votes
1answer
50 views

A number-theoretic random walk on the integers

Suppose a random walker starts at $S_0 = 2$, and walks according to the following transition probabilities: If the walker is on the $n$th prime number $p_n$, she moves to either $p_n + 1$ or ...
1
vote
1answer
27 views

unramified quadratic extensions of 2-adic numbers

i already know how to get the 7 quadratic extensions of $\mathbb{Q}_2$ from hensel's lemma. they are $\mathbb{Q}_2(\sqrt{d})$ for d = -10, -5, -2, -1, 2, 5, 10. question: which of these are ...
0
votes
1answer
53 views

Least pair of numbers having at least $k$ distinct prime factors

Consecutive numbers with less than $k$ prime factors? shows that for every $k$, there is a pair $(n/n+1)$, such that $n$ and $n+1$ both have at least $k$ distinct prime factors. The object is to ...
0
votes
0answers
92 views

ABC triples when C is a power. [on hold]

For any integer $x$ we can apparently always find a sufficiently large $y$ such that there is an $ABC$ triple $A + B = C$ (from the $ABC$ conjecture $rad(ABC) < C$) where $C = x^y$. The minimum ...
2
votes
2answers
98 views

Special representation of a number

How can I check, if a number $n$ can be representated by $$pq+rs$$ where $p,q,r,s$ are pairwise different prime numbers with the same number of digits. For example, $$105153899965560312960 = ...
2
votes
1answer
98 views

Find prime numbers $a, b, c$ such that $a^b+c$ is a prime

Please help me find all $a, b, c \in \mathbb{P}$ such that $a^b+c$ is a prime Example I just can find: $2^3 + 5 = 13 \in \mathbb{P}$
44
votes
1answer
2k views

$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and ...
1
vote
1answer
69 views

Finding all primes $p$ such that $3p+20$ and $p+20 $ are primes

I need to find all primes $p$ such that $3p+20$ and $p+20 $ are primes. The first primes which satisfy the condition are: $3, 11, 17, 23$. I've tried to find the dependence, and found that $p \equiv ...
4
votes
1answer
39 views

Number of primes between $2k$ and $(\sqrt{k}-1)^2$.

I would like to prove the following. Let $\pi$ denote the prime counting function. Then for $k\geq 81$ we have $$ \pi(2k)-\pi\left(\left\lfloor(\sqrt{k}-1)^2\right\rfloor\right)\geq 6. $$ What I ...
4
votes
3answers
739 views

How to appreciate Fermat's last theorem?

I am someone who is not a Maths major, these days (during the summer) I am attracted to Fermat's Last Theorem. I understand that there is no whole number solution to the equation $x^n + y^n = z^n$ for ...
27
votes
12answers
1k views

Computing irrational numbers

I am genuinely curious, how do people compute decimal digits of irrational numbers in general, and $\pi$ or nth roots of integers in particular? How do they reach arbitrary accuracy?
5
votes
2answers
84 views

$3^x + 4^y = 5^z$ [duplicate]

This is an advanced high-school problem. Find all natural $x,y$, and $z$ such that $3^x + 4^y = 5^z$. The only obvious solution I can see is $x=y=z=2$. Are there any other solutions?
1
vote
0answers
23 views

If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements known to hold in general?

(Note: This question has been cross-posted from MO.) Let $\sigma$ be the classical sum-of-divisors function. A number is said to be perfect if $\sigma(N)=2N$. If $q^k n^2$ is an odd perfect number ...
0
votes
0answers
19 views

Is there a classification of perfect squares in the rings of cyclotomic integers?

We know that perfect squares in $\mathbb {Z} $ has the form $n^2, n\in \mathbb {Z} $. More generally, can I use a nice condition to find a solution of Diophantine equations of the form $x^2=c $, $c ...
0
votes
0answers
44 views

Proving the existence of a sequence such that

I am trying to prove that there exists a sequence, for example: $$ f(n) = n! $$ (or we can select any sequence we need to prove the existence of just one), with the following property: edit: for ...
5
votes
3answers
928 views

Where is the mistake in this proof about rationals having eventually periodic decimal expansions?

I know that a rational number has an eventually periodic decimal expansion, and not necessarily just periodic. So what is wrong with this 'proof' that any rational number has a periodic decimal ...
2
votes
2answers
49 views

Proof that $a\mid x, b\mid x, \gcd(a,b)=1 \implies (ab)\mid x$

I need to prove that: $$a\mid x, b\mid x, \gcd(a,b)=1 \implies (ab)\mid x$$ What I thought was: $$a\mid x \implies x = aq_1\\b\mid x\implies x = bq_1$$ Also, since $\gcd(a,b) = 1$, we have that ...