Tagged Questions
9
votes
2answers
50 views
Origin of well-ordering proof of uniqueness in the FToArithmetic
In the Appendix to Ivan Niven's book "Numbers: Rational and Irrational", he proves the Fundamental Theorem of Arithmetic (FToA) without using Euclid's Lemma that if a prime divides a product, then it ...
6
votes
1answer
138 views
Two questions re: $\sum_{n=1}^{\infty}n^{-p_{n}}$
Edit Motivation for question: I looked up the decimal expansion of:
$$\sum _{n=1}^{\infty } \sum _{k=n}^{\infty } k^{-2 k},$$
which matches the first seven digits of the function in question. I would ...
0
votes
3answers
93 views
which texts on number theory do you recommend? [duplicate]
my close friend intend to study number theory
and he asked me if i know a good text on it , so i thought that you guys can help me to help him !
he look for a text for the beginners and for a first ...
2
votes
0answers
72 views
If $r$ is a primitive root of $m$, then $r$ is also a primitive root of each positive divisor $d$ of $m$
If $r$ is a primitive root of $m$, then $r$ is also a primitive root
of each positive divisor $d$ of $m$.
There is a proof of this statement here (see Theorem 4.4), but I cannot find this ...
2
votes
0answers
52 views
Looking for a very gentle first book on number theory
My gf is the classic math-phobe, totally traumatized by math, etc., which surprises me, since she's whip-smart. The only explanation I can think of is that she got off to a bad start in elementary ...
2
votes
1answer
123 views
What is a good book to learn number theory?
What would be a good book to learn basic number theory? If possible a book which also has a collection of practice problems? Thanks.
4
votes
2answers
61 views
Foundation on Diophantine Analysis and Number Theory
I want to read particularly about diophantine Analysis and Elementary Number Theory from a novice level.
The books which I found on net:
A Guide to Elementary Number Theory by Underwood Dudley
...
3
votes
0answers
70 views
$M_n=2^n-1$ Mersenne numbers in mathematics
Did the Mersenne numbers turn out to be interesting in other fields of mathematics besides the Numbers Theory?
In other words, the function $M(n)=M_n$
where
$$M_n=2^n-1$$
or the recursive realtion
...
13
votes
2answers
234 views
How did Letac solve $x_1^k + x_2^k + \dots +x_9^k = 0$ for $k = 1, 3, 5, 7$ in 1942?
It's quite easy to find integer solutions to,
$$x_0^k + x_1^k + \dots +x_9^k = 0$$
for $k = 1, 3, 5, 7$. One I found is, if $x^2-10y^2 = 9$, then,
$$1 + 5^k + (3+2y)^k + (3-2y)^k + (-3+3y)^k + ...
0
votes
2answers
88 views
The sum of a square and twice a square
Consider the set $\{3,6,9,11,12,17,18,19,22\ldots\}$ (OEIS A154777) of positive integers that are expressible in the form $a^2 + 2b^2$. Is there a theorem about the form of such numbers analogous to ...
5
votes
1answer
150 views
Modified Euler's Totient function for counting constellations in reduced residue systems
I am working on a modified totient function for counting constellations in reduced residue systems for the same range that Euler's totient function is defined over. This post is separated into three ...
1
vote
0answers
41 views
Are there (known) bounds to the following arithmetic / number-theoretic expression?
I apologize in advance if this is something that is already well-known in the literature, but I would like to ask nonetheless (for the benefit of those who likewise do not know):
Are there (known) ...
4
votes
2answers
97 views
If $A\equiv 1\pmod{3}$, then $4p=A^2+27B^2$ uniquely determines $A$.
If $p\equiv 1\pmod{3}$, it's well know that $p$ can be expressed as
$$
p=\frac{1}{4}(A^2+27B^2).
$$
In this letter by Von Neumann, he mentions that Kummer determined that $A$ is in fact uniquely ...
6
votes
3answers
180 views
Good introductory readings to topics related to prime numbers for non-mathematicians
I'm a maths hobbyist who is fascinated by prime numbers. My quest to delve into the interesting parts of the topic is always hindered by my inability to understand the notation and concepts I ...
0
votes
3answers
63 views
Need a recommendation for a book that covers congruence classes.
I want to learn about congruence classes and need a book recommendation. With a lot of examples.
3
votes
1answer
127 views
Reference request: Clean proof of Fermat's last theorem for $n=3$.
I have seen a proof for FLT, $n=3$ using factorisation in the ring of Eisenstein integers, but it's quite long and convoluted; I am wondering if there is a more 'advanced' proof which avoids infinite ...
7
votes
1answer
59 views
Parameters giving maximal-length Collatz-like sequence
In a recent question the following recursive sequence was considered:
$$
a_{n+1} = \cases{\frac{a_n}{2} & $a_n$ is even \\ a_n +d & $a_n$ is odd}, \quad a_1 = d + 1
$$
where $d$ is an odd ...
4
votes
2answers
133 views
Solving simple congruences by hand
When I am faced with a simple linear congruence such as
$$9x \equiv 7 \pmod{13}$$
and I am working without any calculating aid handy, I tend to do something like the following:
"Notice" that adding ...
17
votes
2answers
273 views
Multiplication tables with all entries distinct
Let positive integers $\alpha$ and $\beta$ be given. It is easy to
find sets $A$ and $B$ of positive integers such that:
$|A|=\alpha$ and $|B|=\beta$
The set $P = \{ab : a\in A, b\in B\}$ contains ...
8
votes
1answer
108 views
Summing over a cyclic subgroup of a multiplicative group mod n
Let $x$ be a unit in $\mathbb Z/ n \mathbb Z$ of multiplicative order $m$.
I am trying to determine when it is that
$$
\sum_{i=0}^{m-1} x^i \equiv 0 \mod n .
$$
Is this kind of situation something ...
10
votes
1answer
173 views
Diophantine equation $x^y-y^x=11$
How can one find all integer solutions to $x^y-y^x=k$, for a given k?
Example case $x^y-y^x=11$
0
votes
0answers
85 views
Literature for this quasi-proof of the twin prime conjecture
What follows is a quasi-proof that I desire additional reference material for. In my lack of lemmata to cite, I have chosen to write this out in prose in the hopes that someone may point me to the ...
1
vote
1answer
142 views
When is the concatenation of numbers a square?
A while back I asked a question concerning the existence of squares of the form $1234567891011\ldots$ This question is similar but more general. Let $n,p,\ell$ be positive integers, then we define ...
1
vote
3answers
115 views
When does a prime $q$ not divide $\lambda p +1$ for all $\lambda$?
What can be said about the following question?
For which prime numbers $p$ there exists another prime number $q(p)$ such that $q(p)$ does not divide $\lambda p +1$ for all integers $\lambda$ ?
For ...
3
votes
4answers
175 views
Number of quadratic residues mod $p$
I saw in a comment to this question that there are exactly $\frac{p-1}{2}$ quadratic redidues in $\mathbb{F}_p$, but I cannot find the proof by myself (it's been ages since I last touched this kind of ...
1
vote
0answers
78 views
Polygonal numbers modulo odd primes
Marc Renault's masters thesis "Properties of the Fibonacci Sequence Under Various Moduli" is well known for its investigation of Fibonacci numbers with focus on the distribution of residues, peiods of ...
2
votes
2answers
237 views
Primitive roots of odd primes
The following facts about primitive roots of an odd prime seem to be well known. For example, they both appear as exercises in Burton's Elementary Number Theory.
Let $p$ be an odd prime. Then:
...
1
vote
2answers
338 views
Discussion on twin prime conjecture
I understand where I am wrong in my previous post. Also, I am very thankful to all members, who answered and showed my errors in post. Now, I would like to know the proof for the following.
"The ...
1
vote
2answers
158 views
Something like : “recursive” harmonic numbers? Where can I read more?
In my other thread I discussed a matrix-decomposition; for one matrix (U) I found now a description of its entries, which may best be denoted as "recursive harmonic numbers". However, googling with ...
1
vote
1answer
227 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
89 views
Simple Answer to Showing $N_{k/\mathbb{Q}}(\alpha)=N((\alpha))$
So I'm trying to show that if we have some number field $k/\mathbb{Q}$ and ring of integers $R_k\subset k$, and an element of $R_k$, say $\alpha$, that the field norm of $\alpha$ is equal to the ...
3
votes
1answer
57 views
Collection of congruencies
I have been going around various questions based on number theory in this forum, and what I have found is that congruencies serve as an important tool in many of the questions and actually simplify ...
4
votes
1answer
408 views
Reference request for the following proof of Euclid's Lemma
I'm looking for a reference containing the following proof of Euclid's Lemma.
Recall the statement: Let $a,b$ be positive integers and let $p$ be a prime dividing $ab$. Then $p$ divides $a$ or $b$.
...
4
votes
1answer
219 views
Why is $a\equiv b \pmod n$ equivalent to the congruences $a\equiv b,b+n,b+2n,\dots,b+(c-1)n\pmod {cn}$?
I learned the following proposition (in which there is no proof) in a GRE math preparation book. I don't understand what it means and I am not able to find any theorem about this statement in Hardy's ...
2
votes
1answer
340 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 ...
8
votes
4answers
978 views
Looking to understand the rationale for money denomination
Money is typically denominated in a way that allows for a greedy algorithm when computing a given amount $s$ as a sum of denominations $d_i$ of coins or bills:
$$
s = \sum_{i=1}^k n_i ...
