0
votes
1answer
28 views

divisibility on prime and expression

This site is amazing and got good answer. This is my last one. If $4|(p-3)$ for some prime $p$, then $p|(x^2-2x+4)$. can you justify my statement? High regards to one and all.
3
votes
0answers
24 views

how to find the last non-zero digit of $n$

I want to know how to find the last non-zero digit of $n$. For example $n = 100!$ my try: First i have to know how much Zeros $100!$ has so i did this: $$E_{5}10 = \sum _{1\leq k <n} ...
1
vote
0answers
48 views

Count ways to reach Nth row

Given a N*M grid I need to reach last row with following operations : ...
3
votes
2answers
62 views

partitions and their generating functions and Partitions of n

A partition of an integer, n, is one way of writing n as the sum of positive integers where the order of the addends (terms being added) does not matter. p(n, k) = number of partitions of n with k ...
0
votes
1answer
28 views

Hilberts Theorem (norm group)

The theorem says the following: The map $N$ is a group homomorphisim from the multiplicative group of $\mathbb{Q}^{x}[i]$ to the multiplicative group of $\mathbb{Q}^{x}$ and has kernel $\lbrace ...
0
votes
2answers
55 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, ...
0
votes
1answer
28 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 $$\pi (x) \leq \frac{x}{2}+1$$
1
vote
0answers
55 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 ...
0
votes
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.
6
votes
4answers
854 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 ...
0
votes
4answers
89 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 (i.e. in the form $2^{2^n} + 1, n \in \mathbb N$) 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 ...
1
vote
1answer
47 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) = ...
1
vote
1answer
40 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 ...
1
vote
1answer
30 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 ...
2
votes
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 & & & ...
0
votes
3answers
35 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 $d|4m^2+5m+4$ ...
13
votes
6answers
976 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 ...
3
votes
4answers
52 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 ?
-1
votes
1answer
91 views

Cut squares from sheet

A rectangular paper sheet of M*N is to be cut down into squares. ...
1
vote
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 ...
0
votes
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 ...
0
votes
0answers
18 views

Quickie on NT notation

Is there a notation for the set of quadratic residues of an arbitrary natural $n$? I can't seem to find it anywhere on the internet, and it would be very nice if I could use this instead of every time ...
1
vote
2answers
50 views

Lucas Numbers Proof $L_n = \alpha^n + \beta^n$

Proof by Induction: Lucas numbers are recursively defined as: $L_n = L_{n-1} + L_{n-2}$ where $L_1 = 1$ and $ L_2 = 3 $for $n \ge 3$ Show that: $L_n = \alpha^n + \beta^n$ for $\alpha = ...
2
votes
1answer
51 views

Show that the equality is true

If $f$ is a Completely multiplicative function and $g$ is an arithmetic function such as $g(1) \neq 0$ prove that: $$(f\cdot g)^{-1} = f\cdot g^{-1}$$ Any function with the -1 as exponent is the ...
2
votes
1answer
33 views

How to show this equality

If $f$ is a multiplicative function and ¨$n$¨ is a square-free positive integer. Prove that: $$f^{-1}(n) = \lambda(n)\cdot f(n)$$ where $f^{-1}$ is the dirichlet inverse and $\lambda$ is the ...
0
votes
1answer
27 views

Form of solutions of Diophantine equation

Consider the following Diophantine equation $$zx^2 +xy^2 +yz^2 =xyzt .$$ Is that true that all solutions of this equation are of the form $(x,y,z,t) =(a^2 b ,b^2 c ,c^2 a ,t)$ for some ...
2
votes
2answers
61 views

How to show that [closed]

If $f:\mathbb{Z}\longrightarrow \mathbb{N}$ such that $\forall x,y \in \mathbb{Z}; f(x+y)\cdot f(x-y) = [f(x)\cdot f(y)]^{2}$ Prove that $log_{f(1)} f(z)$ is a perfect square $\forall z \in ...
1
vote
0answers
39 views

sum over primes involving divisor function (variation of the Titchmarsh divisor problem)

Does there exist an asymptotic estimate for the following sum over primes $$ \sum_{p\leq x} \frac{\tau(p-1)}{p}\;, $$ where $\tau(n)=\sum_{d|n}1$ is the divisor function?
2
votes
1answer
32 views

p-adic numbers and GCD

Given two numbers $a,b \in \mathbb{Z}$, how do we prove that the $p$-adic number of $\gcd(a,b)$ is the same as the minimum for the $p$-adic number of $a$ and the $p$-adic number of $b$? Does this ...
3
votes
1answer
48 views

Foundational proof for Mersenne primes

I know how to prove that, if $2^n-1$ is prime and $n>1$, then $n$ is prime. But how do we prove that, if $a^n-1$ is prime and $n>1$, then $a$ must equal 2?
2
votes
2answers
25 views

Question about G.C.D.

Let, $$a_{n}=n^2+20$$ $$d_{n}=\gcd(a_{n},a_{n+1})$$ where $n$ is a positive integer. Find the set of all values attained by $d_{n}$ I tried, $d_{n}=\gcd(n^2+2n+21,n^2+20)$ ...
0
votes
0answers
68 views

How does $\sum\limits_{i=1}^n i^c \lfloor{n/i}\rfloor$ converge?

How to converge such series : $$y = \sum\limits_{i=1}^n i^c \lfloor{n/i}\rfloor$$ where c can be any constant value, and particularly $i^c = f(i).~$ Also, $f(x)$ is not a Euler's totient function.
0
votes
0answers
24 views

Similar to Bernoulli sequence

The sequence of Bernoulli numbers $(B_m)_{m\ge 0}$ can be defined by $B_0=1$, $B_1=-1/2$ and $\sum_{i=0}^{n-1}{\binom{n}{i}B_i}=0$ whenever $n\ge 2$. This is in turn equivalent to ...
-3
votes
2answers
41 views

How to show that [closed]

If $f$ is a function $f: \mathbb{N}\longrightarrow\mathbb{N}$ that verify the conditions $\forall m,n \in \mathbb{N}:$ i) $f(m+n) = f(m) +f(n) -1$ ii) $f(2n+1) = f(n) + 2(n+1)$ Find $f(2014)$
0
votes
0answers
19 views

Could you give a definition of what is a superior highly composite number using only words?

I know very well what is a superior highly composite number, but I would like to see how could we (roughly) define what is a superior highly composite number using only words (using no equations and ...
1
vote
0answers
72 views

Any Computational Number Theory Book, include software programs for key steps of the proofs of major theorem?

All: Can anyone recommend some Computational Number Theory Books, which include software programs for key steps of the proofs of major theorem ? Some computational number theory books only include ...
0
votes
0answers
12 views

Any work on the alternating sign version of the Dirichlet Divisor Problem?

Suppose I have functions $$\displaystyle E_1(k) = \sum_{j=1}^n (-1)^{j+1}$$ $$\displaystyle E_2(k) = \sum_{j=1}^n \sum_{k=1}^{\lfloor \frac{n}{j} \rfloor} (-1)^{j+1}(-1)^{k+1}$$ $$\displaystyle ...
0
votes
0answers
31 views

Robin's Inequality

Accoriding to this publication: On Robin’s criterion for the Riemann hypothesis. YoungJu CHOIE, Nicolas LICHIARDOPOL, Pieter MOREE, Patrick SOL On page 2 it states that 2 and 1 fails Robin's ...
0
votes
2answers
63 views

Proof of Fermat's Little Theorem

I just learned about primitive roots today, and then I thought of this proof of Fermat's Little Theorem. Seeing that most proofs of this theorem aren't simple, I think I'm either completely wrong in ...
1
vote
1answer
39 views

How to show that for $Q>1$, $\sum\frac{1}{n}=\frac{\phi(Q)}{Q}\log x + O(1)$, where $n<x$ and $(n,Q)=1$ and where the O constant may depend on $Q$?

Here $\phi$ denotes the totient function. I can't quite get the solution despite trying to use techniques like inclusion and exclusion principle. Help is appreciated.
0
votes
1answer
21 views

Euclidean algorithm for two poloynomials

Find the $gcd(x^8, x^6+x^4+x^2+x+1)$ using the euclidean algorithm. $x^8 = (x^2)(x^6+x^4+x^2+x+1)+(-x^6-x^4-x^3-x^2)$ $(x^6+x^4+x^2+x+1)=(-1)(-x^6-x^4-x^3-x^2)+(-x^3+x+1)$ ...
1
vote
1answer
30 views

Given integers $a$ and $b > 0$, show that there exists a unique integer -

Given integers $a$ and $b > 0$, show that there exists a unique integer $r$ with $0\le r\lt b$ satisfying $a = \left\lfloor \dfrac{a}{b} \right\rfloor b + r$ I am familiar with the Euclidean ...
0
votes
0answers
22 views

Euclidean Algorithm as Linear Combination of Two Sequences

The Euclidean Algorithm is as follows: $r_{i-2} = q_i r_{i-1} + r_i$ where r_{-2}=a and r_{-1}=b. The gcd(a,b) is $r_n.$ Define 2 new sequences as follows: $S_i = S_{i-2} -q_{i}S_{i-1}$ where $S_{-2} ...
2
votes
2answers
190 views

Solutions of Diophantine equation

Does there exists any other solutions of the following Diophantine equation $$zx^2 +xy^2 +yz^2 =xyzt .$$ I found that $$(x,y,z,t) =(s,s,s,3) ,(x,y,z,t)=(s,2s,4s ,5)$$ where $s\in\mathbb{N}$ are ...
1
vote
1answer
35 views

Multiplicative submonoid: an exercise

Here's my problem: (a) Show that the set of integers, which can be written as $a^2+ab+b^2$ for some $a,b \in \mathbb{Z}$ is a multiplicative submonoid of $\mathbb{Z}$; (b) Explain how all ...
0
votes
2answers
69 views

Check my proof of Lehmers conjecture [closed]

$\phi{(n)}=n-1$ for $n$ being composite. Here, $\phi{(n)}$ represents the Euler totient function. (1-1/p1)(1-1/p2)......(1-1/pn)=((n-1)/n) because this will prove that Phi of n=(n-1).. We need to ...
0
votes
4answers
66 views

Solve $c^2-b^2-a^2=2N$

Is there anyone that can help solving this equation: $c^2-b^2-a^2=2N$ where $a,b,c,N$ are natural numbers. Edit: We need to express $a,b,c$ for a certain $N$. Regards
2
votes
1answer
53 views

Count ways to make total coin value [closed]

For any non-negative integer K, suppose we have exactly two coins of value 2^K (i.e., two to the power of K). Now we are given a long N. We need to find the number of different ways we can represent ...
6
votes
3answers
93 views

Does the sum of the reciprocals of all primes of the form $4k+1$ converge?

Let $S=\{p\in \mathbb{Z}^+ : p\ \text{is prime and}\ p\equiv 1 \mod \ 4\}.$ Is $\displaystyle\sum_{p\in S}\frac{1}{p}$ finite or infinite, and where can I find more information about it?
3
votes
2answers
82 views

Finding $23! 7! \bmod 29$ using Wilson's Theorem

I'm trying to reduce $23!\,7! \bmod 29$. I used Wilson's Theorem to get $23!(120)\equiv 1 \pmod{29}$. I then solved $120a\equiv 1 \pmod{29}$ and got $a\equiv 22$. I then computed $7! \pmod {29}$. ...