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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5
votes
1answer
106 views

Is it possible to find a perfect cube like 111…11?

Can we find a perfect cube like $111...111$(all digits are $1$), apart from the number $1$ itself? It's easy to prove that there can't be anything like $111...11$ that is a perfect square besides ...
26
votes
1answer
379 views

Are there $n$ groups of order $n$ for some $n>1$?

Denote $N(n)$ : the number of groups with order $n$. Can $N(n)=n$ hold for some $n>1$ ? I checked the OEIS-sequence as well as the squarefree numbers $n$ in the range $[2,10^6]$ and found no ...
-5
votes
0answers
67 views

Simple logical proof of Fermat's Last Theorem [closed]

My interest in the Fermat Conjecture (FC,) began as an interest in the Pythagorean theorem. I wasn't looking for integer solutions of n>2. I was more interested in the fact that odd integer values of ...
1
vote
0answers
12 views

$n^a$ integral for all integer $n$ implies $a$ integral

Let $a>0$ be a real number, such that for all integers $n\geq 1$: $n^a \in \mathbb N$ Show that $a$ must be an integer. It's not difficult to show this when $a$ is a rational number: ...
2
votes
0answers
12 views

Finite amount of consecutive smooth numbers

is there a short proof of the fact that there is a finite amount of consecutive smooth numbers (meaning Given a finite set B, there is a finite amount of pairs $n,n+1$ so that both can be expressed as ...
0
votes
0answers
13 views

Abstract norm map in Neukirch's book

At page 277 of Neukirch's ANT is defined the abstract norm map and the norm residue group, but I have a problem that I'll explain below: $G$ is a profinite group and the closed subgroups of $G$ are ...
0
votes
0answers
13 views

What are some easy to prove results on the density of primes?

Bertrand's postulate states that for any integer $n>3$, there's always a prime $p$ between $n$ and $2n-2$. That result sets a reasonable 'lower bound' on how often we can expect primes to show up, ...
7
votes
6answers
570 views

Squaring both sides when units are different?

Given $((9) \text{inches})^{1/2} = ((0.25) \text{yards})^{1/2}$, then which of the following statements is true? $((3) \text{inches}) = ((0.5) \text{yards})$ $((9) \text{inches}) = ((1.5) ...
0
votes
0answers
13 views

An upper bound for the Chebyshev function?

The Chebyshev functions are defined as $\psi(x) = \sum_{p^m \leq x} \log n$ and $\theta(x) = \sum_{p\leq x} \log p$, where $p$ is a prime, $m\geq 1$ is an integer and $n=p^m$ in $\psi(x)$. It is known ...
0
votes
2answers
32 views

Solve the following congruence: $x(x+1)(x+2) \equiv 0 \pmod{221}$

Find the first five solutions for, $$x(x+1)(x+2) \equiv 0 \pmod{221}$$ I am very confused. By CRT, $x(x+1)(x+2) \equiv 0 \pmod{13}$ and $x(x+1)(x+2) \equiv 0 \pmod{17}$ But these two ...
-3
votes
0answers
26 views

number based problem [on hold]

You have two numbers which is "444444......4"(2016 4's) & "88888.....89"(2015 8's). Now add the two numbers then calculate the root.then calculate the sum of the digits of root.
3
votes
1answer
44 views

Step to prove twin primes' conjecture: $\liminf_{n\to\infty}(p_{n+1}-p_n)<7\cdot10^7$

Today I have found that the Chinese mathematician Yitang Zhang has proven in 2013 that the sequence $d_n=p_{n+1}-p_n$ where $p_n$ is the $n$th prime has a finite inferior limit (and in fact, lesser ...
1
vote
2answers
32 views

Prove that there exists infinitely many primes of Digital root $2,5$ or $8$

I am highly interested in properties of digital root. Digital Root: Digital root of a number is a digit obtained by adding digits of number till a single digit is obtained. It's clear that Digital ...
1
vote
1answer
10 views

order of a subrgoup of rank $r\geq 2$ in $\mathbb{F}_p^*$

Let $a,b\in \mathbb{F}_p^*$ with orders $o_p(a)=|\langle a \rangle|=\alpha$ and $o_p(b)=|\langle b \rangle|=\beta$. I have few questions: 1) Is it true in this case ($\mathbb{F}_p^*$ cyclic) that ...
3
votes
5answers
143 views

How does one explain addition?

What is $1 + 2$? The question may seem dumb but how can one prove the answer? I heard there is a proof but don't know where to find it so please help. Thanks in advance.
0
votes
0answers
23 views

Is the absolute value of the intersection of two functions related to the nontrivial zeros always equal to $\sqrt{2}$?

With $\displaystyle \chi(s)=\pi^{-\frac{s}{2}}\,\Gamma\left(\frac{s}{2}\right)$ and $K(s)=\Psi\left(\frac{s}{2}\right)-\ln(\pi)$, with $\Psi\left(s\right)$ the digamma function, then the Riemann ...
0
votes
4answers
37 views

How can I prove that $\frac{21n -3}{4}$ and $\frac{15n+2}{4}$ are never both integers?

I have converted this to a problem of modular arithmetic. I seek to prove that $21n-3$ and $15n+2$ are never congruent to $0\pmod 4$ for the same value of $n$. I observed that $21n$ is ...
0
votes
0answers
22 views

How can I define $H+K$? [duplicate]

Let be integers 5 and 100, and let be $H=5Z$ and $K=100Z$ subgroups of the additive group $Z$. How can I define the subgroup $H+K$ ? I think $5Z+100Z=5Z$ because mcd(100,5)=5 but I'm not sure that ...
-3
votes
0answers
20 views

Does the Riemann Hypothesis consider mirror symmetry on its non-trivial zeros?

Setting the bottom corners of the square 1 on the center of two intersected circumferences and taking as center of symmetry the center of that intersection, it's possible to project the square 1 ...
0
votes
0answers
41 views
+50

Is it possible to use Möbius inversion on the last equation to get $\mathrm{core}(n)$?

From this question here: Moreover, if multiplicative function $\mathrm{core}(n)$ is defined to map positive integers "$n$" to square-free numbers by reducing the exponents in the prime power ...
1
vote
1answer
20 views

theory number, number of solutions, not prime numbers

I have been troubled by this: $\tau(2^x \times 3^y)=m$ Being $x$, $y$ and $m$ positive integers Then the number of solutions is $\tau(m)$ I already have done the proof for m prime however cant do ...
2
votes
1answer
46 views

proof of chinese remainder theorem $x=a_1M_1y_1+…+a_nM_ny_n$?

I can't understand the proof of Chinese Remainder Theorem let $x ≡ a_1 (\text{mod }m_1 ),$ $x ≡ a_2 (\text{mod }m_2 ),$ · · · $x ≡ a_n (\text{mod }m_n )$ such that $m_1,m_2,...,m_n$ are relatively ...
2
votes
0answers
30 views

Logic problem: “John's safe's passcode'” question from earlier, with more detail [on hold]

The answer and explanations have already been given at its original post (on Facebook) but I'd like to confirm that it is indeed solvable since there are still some parts I don't quite understand. ...
57
votes
1answer
1k views
+500

Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square

Question: For any $a,b\in \mathbb{N}^{+}$, if $a+b$ is a square number, then $f(a)+f(b)$ is also a square number. Find all such functions. My try: It is clear that the function $$f(x)=x$$ ...
1
vote
0answers
24 views

Examples of equations that creates pseudorandom numbers

I just want to know more examples of equations that creates pseudorandom numbers. Right now I only know the Elliptic Curves. $y^2 = x^3 - 3x + b \pmod p$
0
votes
0answers
66 views

Is there a solution for this problem ?? [on hold]

There a man name john , john has a big safe but he forgot the password. he remembered : the password contain 10 distinct numbers If you add a certain digit in front, the aforementioned amount will ...
1
vote
0answers
15 views

Does there exist a $k$ such that for all $n \ge 3$, $\text{gpf}(\lfloor n^{(\log{n})^k} \rfloor) \gt n$?

Does there exist a $k \in \mathbb{R}$ such that for all $n \in \mathbb{N}, n \ge 3$, $\text{gpf}(\lfloor n^{(\log{n})^k} \rfloor) \gt n$, where $\text{gpf}(x)$ is the greatest prime factor of $x$? I ...
0
votes
2answers
39 views

Special case of Pillai's conjecture

Pillai's conjecture is a generalization of Catalan's conjecture. It's say that for fixed positive integers $A, B, C$ the equation $Ax^n - By^m = C$ has only finitely many solutions $(x,y,m,n)$ with ...
2
votes
1answer
26 views

Find all elements of multiplicative order 18.

Find all elements of $\mathbb{Z}_{19}^*$ of multiplicative order $18$. I started by using Euler's Theorem and since gcd(18, 19) = 1 it implies that $a^{\phi (19)} \equiv 1 \pmod n$. Which means ...
2
votes
3answers
34 views

Product of two primitive roots $\bmod p$ cannot be a primitive root.

I recently proved that the product of all primitive roots of an odd prime $p$ is $\pm 1$ as an exercise. As a result, I became interested in how few distinct primitive roots need to be multiplied to ...
0
votes
2answers
30 views

$17x+11y \equiv 7 \pmod {29}$ and $13x+10y \equiv 8 \pmod {29}$. What are x and y?

Congruency question: if $17x+11y \equiv 7 \pmod {29}$ and $13x+10y \equiv 8 \pmod {29}$, we need to find $x$ and $y$. There may be more than one answer. Not sure how to go about this; any help ...
10
votes
1answer
79 views

Is there a polynomial such that $F(p)$ is always divisible by a prime greater than $p$?

Is there an integer-valued polynomial $F$ such that for all prime $p$, $F(p)$ is divisible by a prime greater than $p$? For example, $n^2+1$ doesn't work, since $7^2+1 = 2 \cdot 5^2$. I can see that ...
0
votes
3answers
41 views

Variation on Fermat Little Theorem

Does the following variation of Fermat Little Theorem hold? How do you prove it? Let $p$ be a prime number greater than $3$. Then there exist a natural non-prime $m > 1$ such that ...
0
votes
0answers
21 views

eigenbasis of old forms

Let $S_k(N)$ be the space of weight $k$ cusp forms with respect to $\Gamma_0(N)$ for $1\leq N\leq 100$. We have a decomposition: $$S_k(N)= S_k^{\text{new}}\oplus S_k^{\text{old}}$$ Suppose that we ...
-1
votes
0answers
14 views

On the upper bound for the Chebyshev function: What am i missing here?

The Chebyshev second function is defined as $\psi(x) = \sum_{p^m \leq x} \log p$, where $p$ is a prime, $m\geq 1$ is an integer and $n=p^m$. It is known that there exist positive constants $c_1$ and ...
2
votes
1answer
42 views
+50

Seeking more information regarding the “hybriation function.”

Definition 0. Given a pair of finite sets $Y$ and $X$, write $Y_X$ for the set of all collections $\mathcal{K}$ of functions $f : Y \leftarrow X$ that are closed under "hybridization", by which I ...
1
vote
0answers
19 views

Denominators of harmonic numbers: asymptotic behaviour.

About the sequence $d_n$ of the denominators of harmonic numbers, I know these facts: It is unbounded, since $p\mid d_p$ for any prime $p$. It contains only one $1$. What more is known? Specially, ...
1
vote
2answers
64 views

Proof that $(2^m - 1, 2^n - 1) = 2^d - 1$ where $d = (m,n)$ [duplicate]

$(2^m - 1, 2^n - 1) = 2^d - 1$ where $d = (m,n)$ my work: I assumed $m = da$ , $n = db$ for $a,b \in \mathbb{Z}$. Now, $2^m - 1$ = $2^{da} - 1$ = $(2^d)^a - 1$ = $x^a - 1$ where $x = 2^d$. similarly ...
2
votes
1answer
56 views

Prove that $\left(\frac{-1}{p}\right)\left(\frac{3}{p}\right)=\left(\frac{p}{3}\right)$

Why does $\left(\frac{-1}{p}\right)\left(\frac{3}{p}\right)$ equal $\left(\frac{p}{3}\right)$?
2
votes
0answers
56 views

Inequalities for the sum of all digits [closed]

Let $S(n)$ denotes the sum of all digits of the positive integer $n$ a) Prove, that $S(n) \le 5 \cdot S(2n)$ for every $n$. b) Prove, that there exist infinitely many $n$, with $S(n)>1996 \cdot ...
2
votes
1answer
25 views

Complexity of generating a prime larger than $N$

Is it provably difficult to generate a prime larger than a prescribed $N$? For instance, if I want a prime of $1000$ digits, is there a way to do that deterministically, i.e., without resorting to AKS ...
3
votes
1answer
56 views

Show that the elements of the form $1+\zeta + \zeta^2 + \dots + \zeta^m$, with $1 \leq m \leq p-2$, are invertible in $\mathbb{Z}[\zeta]$

Let $\zeta = e^\frac{2 \pi i}{p}$, with $p$ prime. Show that the elements of the form $1+\zeta +\zeta^2 + \dots + \zeta^m$, with $1 \leq m \leq p-2$, are invertible in $\mathbb{Z}[\zeta]$. I know ...
2
votes
3answers
115 views

How do We know We can Always Prove a Conjecture?

The question asked here is, suppose we are given a a conjecture to prove in number theory (with numerical evidence showing its true). Say an important well studied conjecture that most will believe is ...
6
votes
1answer
103 views

Is $k+p$ prime infinitely many times?

I have the following conjecture: Let $k\in\mathbb{N}$ be even. Now $k+p$ is prime for infinitely many primes $p$. I couldn't find anything on this topic, but I'm sure this has been thought of ...
0
votes
3answers
44 views

Find the value of y in $11y \equiv 14 \pmod{19}$

Find the value of $y$ in $11y \equiv 14 \pmod{19}$. My issue is not with finding a solution. Using the Euclidean algorithm and Bezout's identity I get a final expression of: $$(11)(7)(14) - ...
13
votes
1answer
198 views

CFT via Brauer groups vs via ideles

I am interested in the relationship between the following two versions of CFT: Version 1: (Brauer Group Version) Let $K$ be a number field. One constructs, for every finite place $v$ of $K$, a map ...
7
votes
1answer
101 views

Arithmetic/category theoretic information encoded in $q$-series reciprocals

According to the pentagonal number theorem: $$\prod_{n=1}^{\infty} (1-q^{n}) = \sum_{k=-\infty}^{\infty} (-1)^{k}q^{k(3k-1)/2}$$ Now the reciprocal of this has the partition numbers $p(k)$ in its ...
1
vote
1answer
25 views

Prove that if $17 \not\mid n$, then either $17 \mid n^8+1$ or $17 \mid n^8-1$

Question is : Let $n$ be a natural number not divisible by $17$. Prove that either $n^8+1$ or $n^8-1$ is divisible by $17$. I tried to solve using Fermat theorem for a prime number $p$, and any ...
0
votes
1answer
16 views

Base 7 Conversion with Exponents

Explain how $56.42(4+3)^2=5642$ can be a true statement. I understand that we essentially need $(4+3)^2$ to act as $100$ in order for $56.42$ to become $5642$. However, if we operate in base $7$ and ...
3
votes
0answers
27 views

Does there exist $a\in\mathbb N$, $b\in\mathbb Z$ that $2^na+b$ is a square for all $1\le n\le5$?

We consider such $a\in\mathbb N$, $b\in\mathbb Z$, o numbers of the form $2^na+b$ is square to the largest possible number of values of $n=1,2,3,4,\ldots$. It is easy to see that for $a = 60 $, $ b ...