1
vote
1answer
39 views

Other Interesting solutions to $a=bq+r$? [on hold]

The division algorithm says $a=bq+r$, with $r$ between $0$ and $b$. Are there interesting restrictions on $r$ using number-theoretic properties that make the equation $a=bq+r$ hold, or hold with ...
6
votes
3answers
201 views

Number Theory Reading List

What are the essential number theory texts that every serious student of number theory should read?
5
votes
0answers
109 views

Riemann Zeta function, quaternions and physics

Disclaimer: This question is rather vague, and thus probably not suitable for Mathoverflow, so I prefer to ask it here. I'm sorry if it doesn't meet the standards of this site. Several years ago, I ...
0
votes
1answer
66 views

Numbers that pass every known primality test

Was having fun reminding myself of the inner workings of a few primality tests today and wondered if there exists a composite number that (perhaps provably) passes all known tests? (Ignoring tests ...
4
votes
1answer
132 views

Number Theory: Easy Question, Difficult Answer [closed]

A common impression about number theory is that its questions can sound elementary but their solutions can be extremely difficult. For examples we need to look no further than Fermat's Last Theorem, ...
5
votes
1answer
57 views

Exact rank of Elkies curve

A naïve question. We definitely know an elliptic curve of rank $28$ or more exists by Elkies but no one knows exactly what the rank is for this curve (and for similar examples given previously). ...
20
votes
8answers
2k views

Statements with rare counter-examples [duplicate]

This is a soft question. I'm searching for examples of mathmatical statements (preferably in number theory, but other topics are also fine), that seem to be true, but are actually not. Statements ...
2
votes
2answers
89 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
2answers
72 views

Has this problem been studied?

Today in Italy all students under 18 faced with a test used to establish the quality of the schools they are in. I read one of the question (a mathematical question, of course!) that make me thing for ...
0
votes
1answer
69 views

How to find the smallest prime divisors of $2^{19}-1$ and $2^{37}-1$?

How to find the smallest prime divisors of $2^{19}-1$ and $2^{37}-1$ ? I'm new to elementary number theory and I'm not sure what to do AT ALL. We're currently studying primitive roots and indices.
6
votes
1answer
230 views

What are the active branches of number theory?

Context: I am a junior math major and am hoping to go to grad school after next year for a PhD. I have completed most of the standard undergraduate courses and have been consistently most interested ...
5
votes
2answers
234 views

So can anybody indicate whether it is worthwhile trying to understand what Mochizuki did?

So I am looking at some math stuff and I start looking at the abc-conjecture. Naturally I run into the name Mochizuki and so start trying to see what he did. Well, he is starting look like another ...
1
vote
2answers
139 views

Recommendation for Number Theory Textbook

. Greetings, every mathematicians! I'm a foreigner (meaning English is not my first language) and an undergraduate student. I'm currently studying linear algebra, set theory and have already studied ...
0
votes
1answer
65 views

What class am I most prepared for?

I've only taken up to calc 3, discrete, and linear algebra. Which course am I most prepared for? I'm going to be taking differential equations and advanced calc, but I want to take a 3rd class. I can ...
2
votes
2answers
174 views

Solved Problems in Algebraic Number Theory

I'm not too sure whether this is the right place to ask this (and please correct me if it is not), but I'm currently studying a course in Algebraic Number Theory and would like to be pointed in the ...
3
votes
4answers
87 views

Can anyone prove for every $a,b \in \mathbb Z^+ < p$ ( $p$ is a prime), $p \nmid ab$?

Can anyone prove for every $a,b \in \mathbb Z^+ < p$ ( $p$ is a prime), $p \nmid ab$? I was trying my best to do the problem but like I don't know where to start or anything!
0
votes
0answers
61 views

Heavy Application of Fermat's Theorem

Show that if p = 4k + 3 is a prime, then the product of all the even integers less than p is congruent modulo p to either 1 or -1. I know now that Fermat's theorem implies that 2^((p-1)/(2)) == 1 or ...
1
vote
6answers
863 views

Is this a theorem in Number Theory? I can't find this in my textbook

"If $a \equiv b \pmod m$ and $a \equiv b \pmod n$ and $gcd(m,n)=1$, then $a \equiv b \pmod {mn}$ " Is that a true theorem? I can't find it in my textbook!
1
vote
1answer
43 views

Standing Generalizations of the Collatz Conjecture?

I remember reading that Erdös once that mathematics isn't matre enough to tackle the collatz conjecture. However the Collatz conjecture seems like a rather specific problem to me. Are there any ...
8
votes
1answer
123 views

I made a primality test and want to publish it

So I am a high school student, and I am very interested in maths, and I made my own primality test which also expresses all composite numbers with last digits of 1,3,7, or 9 in just 9 simple ...
0
votes
2answers
62 views

Doing for the first time a mathematical essay! Tips [closed]

I am an undegraduate and i though of doing my first essay on number theory!I kinda have the topic and i started selecting info from different references..but i dont want it to be just a copy paste ...
9
votes
4answers
486 views

Topics on Number theory for undergraduate to do a project

Im an undergraduate in the mathematics field ..So i wanna be alittle more productive and wanted to do an essay or project mostly on number theory or Algebra(Rings or Groups) and i want to ask if you ...
3
votes
2answers
238 views

Existence of algorithm for determining if a given number is rational or not

As far as I understand, it is not necessarily a easy thing to prove that a real number is rational or not. For example, according to http://mathworld.wolfram.com/e.html "$e+\pi \in \mathbb{Q}$?" is ...
4
votes
1answer
269 views

Forcing the discriminant of an integral basis to be a Carmichael number.

I was thinking about the following lemma recently. Lemma: Let $K=\mathbb{Q}(\theta)$ for some algebraic number $\theta$ and let $n=[K:\mathbb{Q}]$. If $\{\tau_1, \,\dots\,, \tau_n\}$ consists of ...
25
votes
4answers
5k views

Why are so many of the oldest unsolved problems in mathematics about number theory?

Stillwell mentions in his book, Mathematics and its History that: Most of the really old unsolved problems in mathematics, in fact, are simple questions about the natural numbers... Have ...
4
votes
1answer
78 views

Prerequisites for reading Automorphic Forms on Adele Groups

I'm interested in reading Gelbart's book "Automorphic Forms on Adele Groups". I have a solid background knowledge at the first-year graduate level (I've passed quals), but I don't know what the ...
4
votes
0answers
138 views

start studying advanced topics in number theory

I am a first year undergrad and have had elementary course in Number Theory which includes only basic introductory topics like: divisibility,gcd-lcm,primes,congruences, number theoretic functions etc. ...
1
vote
0answers
32 views

Schoenfeld's limits & almost primes

If I am correct in my understanding, the Tao-Green theorem employed almost primes as density normalisers. If $N_k(x)$ is the counting function of almost primes, is the study of almost primes 'useful' ...
21
votes
9answers
3k views

Is there an example that a theorem in number theory is useful in another field in mathematics?

I know there are two advanced approaches to number theory. That is, algebraic number theory and analytic number theory. I have heard that algebraic geometry, which generally seems completely different ...
2
votes
1answer
171 views

Quadratic Reciprocity Joke

I found a joke on a site made by user Zev Chonoles on quadratic reciprocity, the joke is as follows: $$\text{Quadratic reciprocity: } \left(\frac{p}{q}\right)=\left(\frac{q}{p}\right), \text{ up to ...
2
votes
0answers
109 views

Prime number represented by spiral

There is this image on 9gag, with description: "Comparison between 5,000 and 50,000 prime numbers plotted in polar coordinates" I thought it might have something to do with Ulam spiral, but ...
4
votes
1answer
169 views

Was Fermat's last theorem proved based on Peano's postulates?

Is the proof of Fermat's last theorem solely based on the Peano's postulates $+$ first order logic? Or it contains other axiomatic systems as well? What does it mean from foundations of math ...
4
votes
4answers
232 views

how many $1$s in the first n digits of $\pi$?

how many $1$s are there in the first n digits of $\pi$? Any good approximation of its distribution? How about the place of the $n$th $1$? Are these two questions related?
0
votes
2answers
70 views

Need a head-start for new topics in maths [closed]

I am a maths enthusiast ( but from electrical engineering background ) . In my college I had two math courses on differential and integral calculus and I had choosen an elective in complex-analysis . ...
8
votes
2answers
84 views

Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
1
vote
1answer
103 views

Relation between the prime density and Riemann's zeros?

Soft question: Where is the connection between the zeros of Riemann's $\zeta$-funciton and the density of prime numbers? Is there a short answer to this question, to get the overview? I once had a ...
6
votes
1answer
85 views

What are the implications of Prime Number Theorem in Cryptography?

I know that primes and prime factorization are the basis concepts in cryptography. However, I would like to know how does the Prime Number Theorem come into picture in cryptography, since it states ...
4
votes
1answer
105 views

What is going on in this degree 8 number field that fails to be a quaternion extension of $\mathbb{Q}$?

This is a soft but very mathematically hands-on question. Hopefully it will be interesting to more than just me. Thanks in advance for your help in thinking clearly about what follows. I have been ...
13
votes
0answers
1k views

What are some strong algebraic number theory PhD programs?

I am currently applying for PhD programs in the US. My main interests are number theory and algebra. More specifically, I am interested in algebraic number theory (number fields, Galois groups, ...
3
votes
1answer
103 views

Real life applications of Maass wave forms

Explaining my work on Maass wave forms to friends and family (all non-mathematician) typically earns me blank faces. So I wonder whether there is some good example to explain their meaning to laymen. ...
3
votes
0answers
112 views

Can one define informational content of a mathematical expression?

At least in physicist's thinking, information, vaguely, is something that allows one to select a subset from a set. Say, a system can be in states A and B, we have done a measurement on it ...
2
votes
5answers
155 views

What's so special about $e$? [duplicate]

If someone with not much mathematics in his luggage asks me: What is so special about $\pi$? then off course I have an answer. Even if $i$ would be the subject (I allready see him gazing at my ...
6
votes
1answer
105 views

Tate's Thesis: Meaning of Local Functional Equation

I am studying the development of Tate's Thesis in Lang's Algebraic Number Theory and have a conceptual question. The setting: Let $k=\mathbb{Q}_p$. Let $\mu$ be the unique Haar measure giving ...
1
vote
2answers
65 views

Does there exist a branch of mathematics that specifically study the number of lattices enclosed by a region?

I have seen that sometimes, in particular in number theory and combinatorial commutative algebra, our questions are somehow related to finding the number of points with integer components in a ...
7
votes
4answers
385 views

Mere coincidence? (prime factors) [closed]

Whether some things in mathematics are mere coincidences might keep philosophers busy for 100,000 aeons, but maybe when such a coincidence gets exploited then it's not a "mere" coincidence any more. ...
1
vote
0answers
411 views

What are real life applications of Diophantine equations?

Are there any real life applications of linear Diophantine equations? I am looking for examples which will motivate students.
8
votes
4answers
361 views

What are applications of number theory in physics?

I was reading Goro Shimura's The Map of My Life. He wrote the following quote in the book. It made me come up with the title question. In particular, is there any application of the theory of modular ...
2
votes
1answer
109 views

The largest number to break a conjecture [duplicate]

There are several conjectures in Mathematics that seem to be true but have not been proved. Of course, as computing power increased, folks have expanded their search for counterexamples ever and ever ...
18
votes
5answers
748 views

Is there a deep reason why $(3, 4, 5)$ is pythagorean? [closed]

The triple $(3, 4, 5)$ is a pythagorean triple - it satisfies $a^2 + b^2 = c^2$ and, equivalently, its components are the lengths of the sides of a right triangle in the Euclidean plane. But of ...
8
votes
4answers
492 views

Connections between number theory and abstract algebra.

I haven't taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat's little theorem, the law of quadratic ...