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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
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
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 …
- 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
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