3
votes
3answers
85 views

How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
0
votes
2answers
72 views

Prove that $(2n+1)+(2n+3)+\dots +(4n-1) = 3n^2$ by induction

Note: This is for self study, the book is Elementary analysis by Kenneth. A. Ross How to prove the following by mathematical induction, I am stuck
1
vote
0answers
35 views

Six digit permutations of 1 to 6 - divisible by 8

I am working on a problem in A Concise Introduction to Pure Mathematics 3rd Ed under the counting and choosing chapter. It is a multipart question and I am stuck on the last part: 'The digits 1 2 3 4 ...
2
votes
2answers
95 views

Second Course in Number Theory - Self Study

I just finished a first course in number theory using Dudley's Elementary Number Theory. This was by far my favorite math course and I want to learn more number theory this summer. As far as ...
2
votes
1answer
61 views

Good introduction to number theory that develops and/or makes heavy use of commutative ring theory and lattice theory?

I'd like to learn some number theory, since it provides a lot of motivation for commutative ring theory and even some motivation for lattice theory (at least, that's the impression I'm under). ...
4
votes
0answers
27 views

Where can I find proof - There're infinitely many primes $p$ such that $p(mod\ N)\not\in H$ - Name?

Origin - http://math.uga.edu/~pete/4400FULL.pdf - on p120, Theorem 122 Fix a positive integer $N>2$, and let $H$ be a proper subgroup of $U(N)=(Z/NZ)^{\times}$. There are infinitely many ...
4
votes
2answers
117 views

There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am ...
14
votes
2answers
1k views

Show that these two numbers have the same number of digits

I want to show that for $n>0$, $2^n$ and $2^n + 1$ have the same number of digits. What I did was I found that the formula for the number of digits of a number $x$ is $\left ...
3
votes
1answer
38 views

Determining quadratic residues quickly

Let's say that I'm looking for all quadratic residues of a number. THe example from my book is 31. So I can just evaluate $i^2\equiv{a}\pmod{31}$, for $i=1..15$. While not a terribly difficult ...
3
votes
0answers
50 views

Prime divisor of the form $2kp+1$ that divides $2^p-1$

The book that I'm reading (Elementary Number Theory by Underwood Dudley) gives a Theorem: If $p$ and $q$ are odd primes and $q|a^p-1$, then either $q|a-1$ or $q=2kp+1$, for some integer $k$. Then it ...
3
votes
0answers
50 views

Totient Function problem

Suppose we know that $(m,n)=2$. Show that this implies that $\phi{(mn)}=2\phi{(m)}\phi{(n)}$. My attempt: So let $m=p_1^{r_1}p_2^{r_2}...p_k^{r_k}, n=p_1^{s_1}p_2^{s_2}...p_k^{s_k}$. Then ...
2
votes
2answers
60 views

system of congruences proof

I've checked a lot of the congruency posts and haven't seen this one yet, so I'm going to ask it. If there is a related one, I'd be happy to see it. Let $x \equiv r\pmod{m}, x \equiv s\pmod{(m+1)}$. ...
5
votes
7answers
348 views

Good Number Theory books to start with?

I'm in Grade 11. I'm interested in elementary number theory and would like properly study it. I'm not intending to enter any competitions.
1
vote
3answers
172 views

Solutions to $x+y+z=31$ and $x+2y+3z=41$

For the equations $$x+y+z=31$$ $$x+2y+3z=41$$ is there a elegant way or method to find all the positive solutions in integers? Thus far, I have been using trial and error (which is time consuming). ...
1
vote
0answers
82 views

is number theory a good place to start learning maths after a long break?

I haven't studied maths since my engineering degree some 15 years ago. At the time, I enjoyed maths and found it quite easy. However, now I feel that I have lost everything I learnt and I'm really ...
1
vote
2answers
68 views

Proof read from “A problem seminar”

May you help me judging the correctness of my proof?: Show that the if $a$ and $b$ are positive integers, then $$\left(a+\frac{1}{2}\right)^n+\left(b+\frac{1}{2}\right)^n$$ is integer for only ...
-2
votes
1answer
116 views

Some elementary number theory questions.

These are few questions I encountered while self teaching some elementary number theory, I've answered some but I don't have any solutions to check If I'm right. If $a \mid m$ and $b \mid n$, prove ...
2
votes
1answer
229 views

GCD to Linear Diophantine Equation without Euclid Algorithm

Is there a technique other than performing Euclid's algorithm in reverse that can elegantly show that if GCD$(a,b) = d$ then there exist integers $x$ and $y$ such that $ax + by = d$?
1
vote
1answer
59 views

Summation of Modulo Sequences

I came across through simulation that multiplying and adding certain modulo sequences yield equal results. Consider the following two sequences \begin{align} g_0[k] &= \sum_{n=0}^{N-1} \left< ...
1
vote
1answer
82 views

Proof for length of period in simple modulo $N$ sequence.

I am looking for a concise proof that the length of the smallest period of the sequence $$f[n] = a n \pmod N $$ is $N$ if $(a,N) = 1$. From the Pigeonhole Principle, it is not hard to show that ...
0
votes
1answer
109 views

Relationship between different sequences generated through modulo arithmetic.

I am unsure the formal mathematical terminology/notation for dealing with sequences generated from integer modulo arithmetic. So first off, could someone recommend a book that focuses on the ...
2
votes
1answer
409 views

Number theory for a high school Calculus student?

I've always loved playing with numbers, but haven't had any formal guidance in the study of advanced mathematics and number theory. Is there a book (or a few books) on mathematics that I wouldn't have ...
2
votes
1answer
497 views

Good book resources (not websites) to learn number theory on my own? [duplicate]

Possible Duplicate: Undergraduate/High-School-Olympiad Level Introductory Number Theory Books For Self-Learning I took number theory this semester and loved it but don't feel like I learned ...