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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1answer
19 views

Number of ordered pairs with given lcm

Suppose $(p,q)$ is an ordered pair of natural numbers with lcm $r^2.s^4.t^2$, where $r,s,t$ are distinct primes.We have to find the number of all such ordered pairs. One simple idea I tried is to fix ...
15
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1answer
110 views

Are there more than 2 digits that occur infinitely often in the decimal expansion of $\sqrt{2}$?

The other day I got to thinking about the decimal expansion of $\sqrt{2}$, and I stumbled upon a somewhat embarrassing problem. There cannot be only one digit that occurs infinitely often in the ...
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2answers
62 views

Prove that $a \sqrt{b + c} + b \sqrt{c + a} + c \sqrt{a + b} \le \sqrt{2(a+b+c)(bc + ac + ab)}$ for $a, b, c > 0$

Prove for $a, b, c > 0$ that $$a \sqrt{b + c} + b \sqrt{c + a} + c \sqrt{a + b} \le \sqrt{2(a+b+c)(bc + ac + ab)}$$ Could you give me some hints on this? I thought that Jensen's inequality might ...
0
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0answers
46 views

If $\gcd(a,b) = \gcd(c,d) = 1$ and $ab = cd$, then $a=c$ and $b=d$. [on hold]

Is this conjecture true? If yes, can somebody help me prove it? If not, can anyone come up with a counter example?
0
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2answers
25 views

Divisibility of Exponents

So I'm having trouble trying to show this, a,b and x are positive integers. If $a\mid b^x$, show that some factor $k$ of $a$ divides $b$. In other words, if a number $a$ divides a power, how can I ...
3
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1answer
112 views

How many pairs $ (a,b)$ of integers such that , $a^2b^2=4a^5+b^3 $

I would appreciate if somebody could help me with the following problem: $Q$: How many pairs $ (a,b)$ of integers such that $$a^2b^2=4a^5+b^3 $$
0
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0answers
27 views

Z is a subset of Q (the set of all rational numbers)

Let $$n \in Z$$ Then $$n*1=n$$ and so $$n=n/1$$ Note n and 1 are both in Z. so n can be written in the form of $$ z = m/n,\,\,\, where\,\, m,n \in Z\,\,\,and\,\,\, n ≠ 0$$so $$n\in Q$$ Is it enough ...
2
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2answers
84 views

Proof that there are infinitely many primes (Euclid)

I was wondering if I could get some insight on my proof. I am in the midst of relearning some number theory and just "writing proofs" in general, and I would like some assistance to see if I am on the ...
0
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1answer
28 views

Solve for bound of $\sigma(n)$ from harmonic series.

I am given the harmonic series, $H_n$. How can I show that for $n\geq 1$, $$\sigma(n)\leq H_n+e^{H_n}\ln{H_n}$$ By the way, if you're not familiar with it, $\sigma(n)$ is the sum of all positive ...
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3answers
51 views

Rational number proposition

**Prop.**Every $$r \in Q$$ can be written as r = m/n, where $$ m,n \in Z$$ such that n>0 and gcd(m,n) = 1 (r is in lowest terms) If I start by saying that let $$r \in Q$$ Then there exist $$a,b \in ...
2
votes
2answers
53 views

What's the value of $x$ in the following equation?

So this is how I approached this question, the above equations could be simplified to : $$a = \frac{4(b+c)}{b+c+4}\tag{1!}$$ $$b = \frac{10(a+c)}{a+c+10}\tag{2}$$ ...
1
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1answer
76 views

Questions on Erdős–Ginzburg–Ziv theorem for primes and understanding related lemmas and their applications.

While trying to prove the prime case of Erdős–Ginzburg–Ziv theorem: Theorem: For every prime number $p$, in any set of $2p-1$ integers, the sum $p$ of them divisible by $p$. I came across with ...
2
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1answer
25 views

Classification of moduli where relatively prime numbers squared are 1

I came across an interesting property of certain numbers with respect to modular arithmetic and I was wondering if anybody had any more information about them. Consider an integer $n$ such that if ...
1
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1answer
25 views

Conditions for existence of quadratic residue congruent to 1

Under what conditions are we guaranteed an existence of quadratic residue 1 other than squares of 1 and -1. What conditions a number must satisfy to have such residue.
0
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1answer
43 views

Partitions of a number with greatest product

For $n\in\mathbb{N}$ choose $k_1,\dots,k_l\in\mathbb{N}$ so that $\sum_{i=1}^{l}k_i = n$. Set $k = \prod_{i=1}^{l}k_i$. What is the largest $k$ that one can get? Is there an explicit formula? What ...
4
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1answer
93 views

How many divisors does $111…1$ have?

Let $A=\underbrace{11..1}_{2010}$. How many divisors does $111...1$ have? Original problem: Prove that $τ(A)>50$ (or $τ(A)<50$) My work so far: If $\tau(A) -$ the number of divisors ...
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0answers
9 views

Finding the roots with the largest magnitude

Given a non-constant polynomial $p\in\mathbb{Z}[x]=\alpha\prod_{k=1}^nx-\alpha_k$ how can I find the roots $R=\{\beta_1,\ldots,\beta_t\}\subseteq\{\alpha_1,\ldots,\alpha_n\}\subseteq\mathbb{C}$ with ...
1
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2answers
47 views

arithmetic mean of smallest numbers of all subsets of r elements formed out of (1,2,..n)

Consider all subsets of r elements of the set $\{1,2,3,......,n\}$ where $1 \leq r \leq n$. Each of these subsets has a smallest member. Let $F(n,r)$ denote the arithmetic mean of these smallest ...
3
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3answers
325 views

Defining the integers and rationals

What axiomatic system is most commonly used by modern mathematicians to describe the properties of the integers and the rationals? Properties like $a+0=a$, $a*1=a$, $a+b=b+a$, Also given these ...
2
votes
1answer
230 views

A question on approximating irrational numbers by rational numbers.

It is known from the theory of continued fractions that if $\epsilon<1/2$ then the only $a,b\in\mathbb{Z}$ such that ...
1
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1answer
19 views

Multiplicative Inverse Modulo N.

I would like to be able to figure out the multiplicative inverse of some integer modulo some N. For example, how would I find ${15^{ - 1}}$ modulo 34. Will these always exist? If not, what dictates ...
3
votes
3answers
272 views

Which Digit-Permutations Preserve Divisibility?

This is a completely random question that just happened to come to mind recently and I was wondering if the MathSE community had anything to say about it. Let $n > 1,b > 1$ be integers and ...
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1answer
136 views

Solving a Word Problem relating to factorisation [closed]

The $\text{Ionof}$ (Integer on number of factors) of an integer is the integer divided by the number of factors it has. For example, $18$ has $6$ factors so $\text{Ionof}(18) = \frac{18}{6} = 3$, and ...
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0answers
39 views

Find the value of $n$

Let $F$ be a field having $5^n$ elements .Also $F$ has an element which satisfies $x^{5^n}=1$ such that $x\neq 1$. Find $n$ . My try: Let $x\in F $ satisfy $x^{5^n}=1$ .Obviously the group ...
0
votes
1answer
34 views

How to find scaling to get minimum positive integer proportion?

Suppose we have x is a strictly positive vector and y=b*x where b is a positive scaling scalar. The problem is to find the function to get the scaling factor b such that y becomes minimum positive ...
21
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4answers
3k views

The equation $x^3 + y^3 = z^3$ has no integer solutions - A short proof

Can someone provide the proof of the special case of Fermat's Last Theorem for $n=3$, i.e., that $$ x^3 + y^3 = z^3, $$ has no positive integer solutions, as briefly as possible? I have seen some ...
-1
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1answer
25 views

What is the bank need to get the message?

In Number theory $p=37, q= 43$, $\phi(pq)= 36 \cdot 42$, $e=5$ $d=?$ What does the bank need to get the message? I don't understand this problem. Can any one help me please?
2
votes
1answer
26 views

How to find a square root mod $pq$ given that $p \equiv q \equiv 3 \pmod 4$

Let $n = pq$ where $p$ and $q$ are prime. We do not know $p$ and $q$. All we know is that $p \equiv q \equiv 3 \pmod 4$. From this we need to find a number $y$, in terms of $n$ and $x$, such that $y^2 ...
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0answers
25 views

Generating functions, Schur's identity

Let $S=\{n\in \mathbb{Z}_+ \mid n \equiv 1, 5 \,\,(\text{mod 6})\}.$ Let $a(n)$ be the number of partitions of $n$ into parts belonging to $S,$ and $b(n)$ be the number of partitions of $n$ into ...
11
votes
3answers
280 views

Show that $(n!)^{(n-1)!}$ divides $(n!)!$

Show that $(n!)^{(n-1)!}$ divides $(n!)!$ I found this question in a text he was reading about BREAKDOWN OF PRIME FACTORS IN FACT, I decided many years, have posted some here to help, but this ...
3
votes
1answer
66 views

Find $n$ for which $\frac{(n!)^2-(n+2)!}{(n+1)!}$ is an integer

I think there is no integer $n$ for which the above expression is an integer, but I am not sure.
3
votes
1answer
334 views

Determining the Collatz Series as a Tree of $\forall\mathbb{N}$

I'm proposing a proof for the Collatz Conjecture; and should like to take answers in terms of validation or contradiction to the arguments proposed. The conjecture states, where; $$ T(n) = ...
0
votes
1answer
20 views

Puzzled with this number theory/analysis problem

So, I am having this problem, let $N(x,y)$ be the greatest integer which $b^{N(x,y)}|x-y$ where $x,y$ are integers in $\mathbb{Z}$. Assume that $b \geq 2$. Show $d(x,y)=b^{-N(x,y)}$ is a metric. ...
1
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1answer
42 views

Difference between Eulers product and Zeta Function at a finite values

So a very important formula proven by Euler is that is equal to Of course these formulas give you the same value when they reach infinity, but my question is that say $s=1$. What would be the ...
0
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3answers
38 views

why if $a \text{ mod } p = -1$ then $a \text{ mod } p = p-1$

This seems very simple and obvious but I can't prove it. why if $a \text{ mod } p = -1$ then $a \text{ mod } p = p-1$? thank you.
4
votes
1answer
29 views

Sum of $k$ consecutive term divisible by $k+1$

Is there an infinite sequence of positive integers such that for every positive integer $k$ sum of every $k$ consecutive terms is divisible by $k+1$?
0
votes
1answer
30 views

Why if $a = kb + c$ then $a \text{ mod } b = c \text{ mod } b$

Here is a very simple question in number theory that I can't prove it. If $a = kb + c$, then I would like to know why the following is true ($a,b,c,k \in \mathbb{Z}$): $$a \bmod b = c \bmod b$$ And ...
0
votes
1answer
22 views

Pollard's $p-1$ method

I've been reading some notes regarding the Pollard's $p-1$ method1 and I came across an aglorithm that (from the math standpoint) I don't fully understand: Given that $\textbf{a = 2}$ and also in my ...
1
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2answers
21 views

Connection between quadratic residue of a number to its factors'

Is it true that, If $N$ is product of two coprime numbers greater than 1. Quadratic residues of these numbers are quadratic residue of $N$ and vice versa? Can someone point me to a proof or show me if ...
1
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0answers
13 views

Embedding of a global field to local field of characteristic $p$/

Let $F=\mathbb{F}_q(t)$ and consider its completion $F_P$ with respect to an irreducible polynomial $P(t)$, namely the local field associated to the place $P$ (I understand this could be technically ...
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0answers
21 views

Convergence behaviour of Eichler integral

Consiger $g : \mathbb H \to \mathbb C$ a modular form of weight $2-k, k \in \frac{1}{2}\mathbb Z$. Let $z \in \mathbb H$ and consider the following integral: ...
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0answers
28 views

N as sum of k primes [closed]

How can we say if N can be represented as a sum of k prime numbers .If N=10 and k=2 it can be represented as sum of two primes (5+5) .How can we say this for any N and K .
0
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0answers
64 views

How to prove non-decreasing sequence?

Empirically, this sequence is non-decreasing: $$\{\pi(n^2+2n)-2n:n\geq0\},$$ where $\pi(\cdot)$ is the number of primes $\leq(\cdot).$ How would we prove it? Edit I have structured a plan to ...
2
votes
1answer
84 views

Prove that $\prod\limits_{2 < p \leq y}\left(1-\frac{2}{p}\right)\sim\frac{D}{\log ^2 y}$ [duplicate]

I'm writing my bachelor thesis about Brun's sieve method and his theorem. In one proof I found this statement without further explanation. It is important to show that the product doesn't converge ...
0
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0answers
20 views

Which Denominations to use for payroll with no returned change

I want to solve the following problem. It is not a homework. Assume that a company pays payroll to employees every period, the sum of the salaries for period is $T$. The accountant goes to the bank ...
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0answers
84 views

The smallest number that if multiplied by 2 forms a permutation of itself

I am looking for the smallest number larger than $0$ which when multiplied by $2$, forms a permutation of itself. I quickly remembered that the number $142,857$ does that, as well as with all numbers ...
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0answers
11 views

Sum of Roots of Unity With Weighted Exponents

I have the following conjecture that I want to believe has some sort of classical result associated to it, but have yet to find any such evidence. Let $\ell,r\in\mathbb{Z}^+$, and fix ...
4
votes
4answers
52 views

Is the 2011th term odd for this sequence and why so?

$a_n = a_{n-1} \cdot a_{n-2} + n$, $n\ge2$, $a_0 = 1$ and $a_1 = 1$. Is $a_{2011}$ odd. Why so? This is not a homework problem. I am appearing for an exam soon and I am solving sample questions for ...
1
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2answers
65 views

Find all natural numbers $m,n$ such that $m|12n-1$ and $n|12m-1$

Find all natural numbers $m,n$ such that $m|12n-1$ and $n|12m-1$ My progress: $m,n $ must be coprime. Also, look at the expression $12(m+n)-1$ it is divisible by both $m,n$
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0answers
59 views

Where can I share findings research with other mathematicians? [closed]

I have created a very interesting integral representation for the sum $\sum_{j=1}^n\frac{1}{j^k}$. From this formula, one can deduce for example the values of Riemann's Zeta function at the positive ...