Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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12 views

How prove that there are $a,b,c$ such that $a \in A, b \in B, c \in C$ and $a,b,c$ (with approriate order) is a arithmetic sequence?

Let $N=\{ 1,2,3,..., 3n \}$ with $n$ is a positive integer and $A,B,C$ are three arbitrary sets such that $A \cup B \cup C = N, A \cap B = B \cap C = C \cap A = \varnothing, |A| = |B| = |C| = n $. How ...
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27 views

Making 24 with given number N

Initially we have a sequence of n integers: 1, 2, ..., n. In a single step, we can pick two of them, let's denote them a and b, erase them from the sequence, and append to the sequence either a + b, ...
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0answers
23 views

Finding a point on an elliptic curve

I have an elliptic curve with the equation $ y^2 = x^3 + ax + b $ in modulo p, where p is prime. I also have a point G on that curve. How can I find another point that isn't a multiple of G? I ...
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1answer
12 views

expository articles on special values of L functions

While searching for some notes on L functions i have seen the following statement... In mathematics, the study of special values of L-functions is a sub field of number theory devoted to generalizing ...
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3answers
85 views

Sum of the digits

Let $N$ be the greatest number that will divide $1305,4665$ and $6905$, leaving the same remainder in each case. Then what is the sum of the digits in $N$?
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1answer
18 views

Proving an inequality on the sum of $\log$ of primes.

Let $S(x)=\sum_{p\leq x} \ln(p)$ where $\sum_{p\leq x}$ denotes a summation over the positive prime numbers that are $\leq x$ Prove that $\forall n \in \mathbb N, S(2n+1)-S(n+1)\leq ...
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1answer
23 views

How can we show that $\pi (x) \leq \frac{x}{2}+1$?

What is the proof that the prime counting function $\pi (x)$ is such that $$\pi (x) \leq \frac{x}{2}+1$$
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26 views

Partition function proof

I am looking for any online information regarding Hardy and Ramanujan's proof, perhaps the proof itself, that the partition function $p(n)$ is asymptotic to $$\frac{e^{K\sqrt{n}}}{4n\sqrt{3}}$$ where ...
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46 views

Mordell Diophantine: $x^2+11=y^3$

I've been trying to solve the diophantine $$x^2+11=y^3$$ recently but to no avail. I tried the "UFD trick", re-writing as $(x-i\sqrt{11})(x+i\sqrt{11})=y^3$, but it didn't give me all the solutions. I ...
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1answer
54 views

Fermat's Last Equation

Sorry this is an amateur question but I was wondering since Andrew Wiles solved Fermat's Last Theorem what effect does this have any impact on Geometry. Does this prove in a sense Higher Order right ...
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2answers
112 views

When $x^2+6xy+y^2$ a square number?

Find all natural numbers $x$ and $y$ such that $x^2+6xy+y^2$ is a square number. For example, $(x,y)=(2,3)$ or $(x,y)=(3,10)$. Obviously, we can consider $gcd(x,y)=1$.
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55 views

Using properties of $\pi$ in the primes

We all know that $\pi$ and the prime numbers are intricately related, thanks to the work of famous mathematicians like Euler and Riemann. The irrationality of $\pi$ can ultimately be used to prove the ...
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0answers
19 views

Upper bound number of distinct prime factors

I want to prove that if $\omega (n)$ is the number of distinct prime factors of $n$ for $n>2$ we have $\omega (n) \leq \frac{\ln n}{\ln \ln n} + O(\frac{\ln n}{(\ln \ln n)^2})$. How can I do this? ...
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0answers
8 views

Name/properties of a difference of continuants

Suppose that $q_1$, $\ldots$, $q_s$ is a sequence of positive integers. Denote by $[q_1, \ldots, q_s]$ the numerator (in lowest terms) of the rational number represented by the continued fraction $$ ...
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0answers
4 views

Points of a lattice inside square of side $N$

Let $\Lambda\subseteq \mathbb Z^m$ be a full-rank lattice of index $h$. I would like to know an upper bound for the quantity $H_N=|\Lambda\cap [-N,N[^m|$ where $[-N,N[^m=\{(a_1,\dots,a_m)\in \mathbb ...
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0answers
59 views

Prove that $\sqrt{n}$ is irrational [on hold]

Question: Using fundamental theorem of integers and the fact that every natural number that is not prime, prove that $\sqrt{n}$ is irrational unless $n=m^2$ for some $m\in\mathbb N$. Here is how I ...
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0answers
7 views

How to show that $f$ is Completely multiplicative function

If $f$ is a multiplicative function and $(f\cdot \mu^{-1})^{-1}= f\cdot \mu$. Prove that $f$ is a is completely multiplicative function. $\mu$ is the Möbius and $\cdot$ is the simple product.
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4answers
834 views

Can any two irrational numbers NOT of the form (m+A) and (n-A) be added to produce a rational number?

$m$ and $n$ being rational numbers, A being an irrational number. I was wondering if two irrational numbers when added always yield an irrational number. All the counter-examples I could find were of ...
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0answers
42 views

Relation between Galois theory and Fermat primes

I am curious about a possible relation between Galois theory and Fermat primes. There is a general solution to any polynomial equation of degree less than or equal to $4$. The only Fermat primes (of ...
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3answers
52 views

How can we find the smallest number n such that 2^(2^n) + 1 is not a prime.

How can we find the smallest fermat number that is not prime and show that it is indeed not a prime? Yes, when n=5, it is not a prime. How can we find n=5 , other than just factoring the fermat number ...
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0answers
27 views

Punctured Elliptic Curve

I've come across the word "punctured elliptic curve" here and there, but none of the basic texts on the topic (Husemoller, Silverman) define or mention it. What point is removed from the curve (the ...
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1answer
46 views

Sum over divisors of sum over coprimes

Set $n \in \mathbb{N}$ , $n>1$ . Consider the function $\phi_{1}$ as $$\phi_{1}(n)= \sum _{r=1 \atop \gcd(r,n) =1}^{n} r$$ Prove that $$\sum_{d|n} d \cdot \phi_{1}\Big(\frac{n}{d}\Big) = ...
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1answer
39 views

$\sqrt {-6}$ is not prime in $\mathbb{Z}+\mathbb{Z}\sqrt {-6}$

Suppose $\sqrt{-6}|(a+b\sqrt{-6})(c+d\sqrt{-6})$. I need to show that $\sqrt{-6}$ does not divide $(a+b\sqrt{-6})$ and does not divide $(c+d\sqrt{-6})$. I thought you might arrive at some ...
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0answers
33 views

Does a generalization of the Teichmuller-character for non-prime arguments exist?

Rereading an older article on Fermat-quotients in which I'd applied some p-adic-rationale I find now, that my method for the representation of bases $b$ which allow high fermat-quotients ...
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0answers
14 views

2 players cross out numbers from the line [on hold]

There is a line with even number of numbers written on it. Two players cross out this numbers one by one from left or right. At the end they find the sum of numbers they crossed out. The winner is one ...
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1answer
39 views

Prime number theorem lemma: prove that $\psi(x)\sim\pi(x)\log(x)$

I'm trying to follow the proof in Wikipedia that the PNT is equivalent to the assertion $\psi(x)\sim x$, by proving that $\psi(x)\sim\pi(x)\log x$, which it claims is a very simple proof. One ...
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1answer
29 views

Best self study book with answers to selected questions for analytic number theory?

All: Can anyone recommend Best self study book with answers to selected questions for analytic number theory ? If a book have no answers to questions, but if you know if some professors choose the ...
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2answers
31 views

Prove that if $\gcd(a,b)=1$ then $\gcd(a^m, b)=1$

I am using the Euclidean Algorithm (EA) for proof. Let $a>b$ and by EA we have $$ \begin{align} a=q_0 b+r_1 & & & \text{where }0\leq r_{1}<b \\ b=q_1 r_1+r_2 & & & ...
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1answer
24 views

which algebraic number theory book with answers to selected questions for self-study?

All: Can anyone recommend some easy to follow algebraic number theory books with answers (hints) to selected questions for self-study ? If a have no answers to questions, but if you know if some ...
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2answers
53 views

How find this minimum of the $q$,if such $\frac{95}{36}>\frac{p}{q}>\frac{96}{37}$

let $p,q$ is postive integer,and such $$\dfrac{95}{36}>\dfrac{p}{q}>\dfrac{96}{37}$$ Find the minimum of the $q$ maybe can use $$95q>36p$$ and $$37p>96q$$ and then find this minimum of ...
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3answers
34 views

Modular calculus and square

I want to prove that $4m^2+1$ and $4m^2+5m+4$ are coprimes and also $4m^2+1$ and $4k^2+1$ when $k\neq{m}$ and $4m^2+5m+4$ and $4k^2+5k+4$ when $k\neq{m}$. Firstly : Let $d|4m^2+1$ and $d|4m^2+5m+4$ ...
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2answers
36 views

Generating function for number of integer solutions, no computer

How do you solve a Generating function for the number of integer solutions with no computer? Use a generating function to solve the number of integer solutions for $$x_1+x_2+x_3=17$$ Where ...
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0answers
19 views

Number of distinguishable arrangements of the word INDOOROOPILLY with three different conditions

I have the following three questions on a past final exam, I wanted to ask if I have done everything correctly. Thank you! How many distinguishable arrangements are there for the letters of the ...
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11 views

Affine cipher and shift cipher

I have the following question: $$An\;affine\;cipher\;with\;key\;K(0,b)\;is\;equivalent\;to\;a\;shift\;cipher\;explain\;why$$ I don't think this is true, and assume it is a typo, $K(1,b)$ I would ...
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1answer
53 views

Perfect squares formed by two perfect squares like $49$ and $169$.

Let $a$ be a perfect square number whose decimal representation is the concatenation of two perfect squares, for example $49$ (from $4$ and $9$), $169$ (from $16$ and $9$), ... and $4900$, $490000$ ...
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1answer
35 views

How many elements does $\mathcal{P}(A)$ have?

Let $A$ be a set of size fifteen. Let $\mathcal{P}(A)$ denote the power set of $A$, that is the set of all the subsets of $A$. How many elements does $\mathcal{P}(A)$ contain? This is the same as ...
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6answers
967 views

Is there any book/resource which explain the general idea of the proof of Fermat's last theorem?

I look for a book/resource which display the general idea of the proof of Fermat last theorem in a simple manner for the public. I mean, books which is not for mathematicians but for the general ...
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0answers
28 views

Abstract algebra: equivalence relations on vector spaces [on hold]

Let $W$ be a subspace of a vector space $V$ over $\mathbb{R}$. (that is the scalars are assumed to be real numbers ). We say that two vectors $u,v \in V$ are congruent modulo $W$ if $(u-v)\in W$, ...
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1answer
23 views

Question about the Chebyshev Inequality.

Let $p_1 < p_2 <\dots < p_n$ be the $n$ first primes listed in crescent order. Using the Chebyshev Inequality (for $x$ sufficiently large) $$0.92\leq \frac{\pi(x)\log x}{x}\leq 1.11,$$ How ...
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0answers
9 views

Bertrand's Postulate and and Chebyshev Inequality

Let $\theta(x) = \sum_{p\leq x}\log p$ and $\pi(x) = |\{p\leq x:p\text{ is prime}\}|$. Using Abel's formula, one can prof the following $$\pi(x) = \frac{\theta(x)}{\log x} + ...
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4answers
47 views

Are all those numbers coprime?

The values of $4m^2+1$ and $4m^2+4m+5$ for $m\geq{1}$ are (resp.) 5,17,37,... and 13,29,53,... Those numbers seem to be all coprime : how to prove it if it is true, please ?
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Sequence terms being divisible

Here's a question I would like hints for: The sequence ${x_n}$ is defined by $x_{n+2}=6x_{n+1}-9x_{n}$ for $n \geq 0$ where $x_0=3$ and $x_1=18$. What is the smallest $k$ such that $x_k$ is ...
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1answer
86 views

Cut squares from sheet

A rectangular paper sheet of M*N is to be cut down into squares. ...
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1answer
17 views

Relation between two numbers.

Let $0 \lt x \lt 1$ and $0 \lt \delta \lt 1$ two real numbers. Can I always find something like $x\lt c\delta^2 \lt 1$, where $c$ is a constant that doesn't involve $\delta$ ?
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0answers
21 views

Find the reflection point $P$

On the real number line, paint red all points that correspond to points of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer points blue. Find a point $P$ on ...
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3answers
43 views

Congruences in number theory

I am working on a worksheet on number theory and I have to solve the following congruences: $$7^{128}=n\mod 13$$ Find $n$. And $$28x^2=1\mod37$$ How should I solve these congruences? I have no clue ...
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1answer
32 views

$\Im\left ((a+bi)^n\right )=0, a,b \in \mathbb{R}, n \in \mathbb{N^*}$

I am trying to connect $a,b,n$ such that $$\Im\left ((a+bi)^n \right )=0, a,b \in \mathbb{R}, n \in \mathbb{N^*} \mathrm{or } \; \mathbb{R^*}$$ What I tried was write $(a+bi)^n$ as $\sqrt{a^2+b^2} ...
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0answers
13 views

An isogeny from a split algebraic torus

Suppose that there is an isogeny (in the category of commutative algebraic groups) from a split algebraic torus to a semi-abelian variety. Does it follows that this semi-abelian variety is also an ...
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1answer
20 views

Number Systems: Determining when they have closure, identities, inverses, and more.

I have the following $9$ number systems at hand and I am to determine which of them possess a particular property. I am having trouble understanding some of the subtleties between the questions and ...
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1answer
39 views

Length in time to find the longest range of primes between 2 and a 13 million character digit?

I am trying to run a program that tells me how many prime numbers there are in a range of numbers. I run it in intervals of 10,000 to 100,000. How long would the program take to determine all the ...