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Results tagged with nt.number-theory
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user 106551
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
3
votes
Which primes $\mathfrak{p} \in \mathbb{Z}[i]$ can be represented as $\mathfrak{p} = x^2 + 2y...
Just elaborating on the very nice comments and answer by Cherng-tiao Perng. As Noam Elkies said, $$
\mathbb{Q}[i,\sqrt{-2}]=\mathbb{Q}[\exp(2\pi i/8)].
$$
As is well known, the Galois group of the $8$ …
1
vote
1
answer
98
views
Limit of quotients of polynomials at fixed value
Let $0<t<1$ be a parameter. Let $n\in\mathbb{Z}$, $n>0$. For $i\in\{0,1,\dots,2^n-1\}$, we always consider $i$ as having $n$ binary digits (positions $1$ to $n$), putting $0$s if necessary. Let $f(i)$ …
-1
votes
Is the sum $\sum_{d\mid n}\frac1{d+1}$ never integral?
For a given set of primes $Q=\{q_1,\dots,q_k\}$, to each prime $p\not\in Q$ we may associate the lattice
$$
L=L_{q_1,\dots,q_k,p}=\{(a_1,\dots,a_k)\in\mathbb{Z}^k: \prod_{i=1}^kq_i^{a_i}\equiv 1 \bmod …
1
vote
Permutations of squares and finite fields
Let $A_q=\{x^2:x\in\mathbb{F}_q^{*}\}$.
Let $\pi^{\prime}$ be the permutation on $A_q$ defined by $$\pi^{\prime}(a_k)=a_{\pi(k)}.$$
Then
$$
\sum_{k=1}^{(q-1)/2}a_ka_{\pi(k)}=\sum_{a\in A_q}(a\pi^{\pri …