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Results tagged with nt.number-theory
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user 3634
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
11
votes
Collecting proofs that finite multiplicative subgroups of fields are cyclic
Lemma: Let $G$ be a finite abelian group, and let $x\in G$ with maximal order. Then for any other element of $y\in G$, $|y|$ divides $|x|$.
Proof. If not, then there is an element $y\in G$ and a pr …
4
votes
The prime numbers modulo $k$, are not periodic
Suppose the primes are periodic mod $k$, and let $p$ be a prime divisor of $k$. Then there must be infinitely many primes of the form $p + nk$, which is divisible by $p$. Bad news.
EDIT:
More prec …
- 12.5k
3
votes
What arrangement of unit cubes minimizes surface area?
Here are some ideas about the related question: for fixed $k$, how many cubes can you arrange while keeping the surface area $\leq k$?
If the surface of an arrangement contains three or more sides …
- 12.5k
3
votes
Special cases of Dirichlet's theorem
I found this generalization of the "$3 \pmod{4}$" version while teaching number theory a few years ago.
Let $G$ be a proper subgroup of $(\mathbb{Z}/n)^\times$. Then there are infinitely many primes …
- 12.5k