36
votes
Accepted
The period of Fibonacci numbers over finite fields
This maybe too elementary for this site, so if your question is closed, you might try asking on MathStackExchange. Many questions about the period can be answered by using the formula
$$ F_n = (A^n-B^...
- 47.4k
29
votes
Algebraic geometry over the complex numbers, and beyond
Algebraic geometry began over the field of reals. What Apollonius of Perga did would be certainly qualified today as algebraic geometry: he classified real plane curves of second order and studied ...
- 91.8k
24
votes
The period of Fibonacci numbers over finite fields
I won't address your question about how small the period of $\{F_n \bmod p\}$ can be as $p$ grows, but instead ask if the upper bounds on the period can be achieved infinitely often. For consistency I'...
- 50.6k
22
votes
Accepted
Conjecture: The number of points modulo $p$ of certain elliptic curve is $p$ or $p+2$ for $p$ of form $p=27a^2+27a+7$
The conjecture is true. To prove it, we can restrict to $k=1$ as explained in the comments. Let $\chi$ denote a cubic Dirichlet character modulo $p$; this exists as $p\equiv 1\pmod{3}$, and it is ...
- 105k
20
votes
Accepted
(Barely) linearly independent vectors over $\mathbb{Z}/2\mathbb{Z}$
Since every $n$ vectors span a space of dimension $n-1$, the whole set $S$ spans a space of dimension $n-1$, and we can assume that $V$ has dimension $n-1$.
Choose $n-1$ linearly independent vectors $...
- 9,501
19
votes
Accepted
Integer solutions for a simple cubic
There are no other solutions.
As Lucenaposition observed, we are seeking integers $x$ such that
$y(x) := (x^3+1) / (7x-1)$ is an integer. Expanding about $x=\infty$
we find that
$$
343 \, y(x) = 49 x^...
- 79.9k
18
votes
Number of commuting pairs (triples, n-tuples) in GL_n(F_q) (and other groups)?
The probability that two elements of a finite group $G$ commute is $k(G)/|G|$, where $k(G)$ is the number of conjugacy classes of $G$. Hence the number of pairs of commuting elements is $k(G)|G|$. ...
- 11.2k
18
votes
Accepted
Does the equation $x^2+x=a$ have an integer solution?
The answer is yes, and in general if $f(x)\in\mathbb{Z}[x]$ is a monic irreducible polynomial whose Galois group contains a cycle of length $\deg f$ (which is always the case when $\deg f$ is prime, ...
- 105k
17
votes
Accepted
Steinberg representation for sporadic simple groups?
The approach that I have taken to generalizing the Steinberg module to finite groups other than groups of Lie type is that, in general, the object we should consider is a chain complex, rather than ...
- 186
17
votes
Is there an $\mathbb{R}$-valued cohomology theory for varieties over $\mathbb{F}_p$?
Let me advertise a conjectural positive answer to a slightly different question. This actually works not only over $\mathbb F_p$ but over its algebraic closure $\overline{\mathbb F}_p$.
Consider the ...
- 21.3k
17
votes
Polynomials which are functionally equivalent over finite fields
The cardinality of $S_f$ is $p^{ \max(0, d+1-p)}$ because $S_f$ consists of polynomials of the form $f + (x^p-x) g$ with $g$ of degree $\leq d-p$.
The fact that such polynomials lie in $S_f$ follows ...
- 148k
16
votes
Accepted
Linear permutations commuting with $x\rightarrow x^{-1}$
The answer is no: a linear transformation of $F$ which commutes with $\phi$ is an automorphism of $F$.
This is a seemingly inelegant but simple argument.
Any map $\psi: F \rightarrow F$ can be ...
- 4,991
16
votes
How many Lie and associative algebras over a finite field are there?
Bjorn Poonen addresses this question for commutative (associative, unital) algebras in The moduli space of commutative algebras of finite rank; asymptotically we have
$$q^{\frac{2}{27} n^3 + O(n^{8/3})...
- 118k
16
votes
Accepted
Existence of a polynomial $Q$ of degree $\geq (p-1)/4$ in $\mathbb F_p[x]$ such that $QQ'$ factorizes into distinct linear factors
Such polynomials always exist, examples are Dickson polynomials of the first kind (with parameter $1$). For a positive integer $n$ these degree $n$ polynomials $D_n$ are most conveniently defined ...
- 22.5k
16
votes
Accepted
The maximal subset of a finite field where the sum of any subset is non-zero
I could trace this question down to the paper of Erdős and Heilbronn "On the addition of residue classes mod $p$" (Acta Arith. 17 (1970), 227–229), where it is shown that for a prime $p$, if ...
- 23k
14
votes
Number of commuting pairs (triples, n-tuples) in GL_n(F_q) (and other groups)?
The number of commuting pairs in any finite group $G$ is well-known to be $k(G)\lvert G\rvert$ where $k(G)$ is the number of conjugacy classes of $G$. There are also well-known generating functions ...
- 44.4k
14
votes
Accepted
Distribution of primitive roots, as p varies
Let me adjust your quantity slightly as
$$\tilde D_p(f) := \frac{1}{\phi(p-1)}\sum_{x \in \Phi(p)} f \left( \frac{x}{p} \right),$$
and let me initially impose the natural condition $f(0)=f(1)$. Then I ...
- 105k
14
votes
Roots of lacunary polynomials over a finite field
Note that the number of distinct non-zero roots of a polynomial $P(x)$ over $\mathbb{F}_q$ always equals to the minimal degree of a non-zero polynomial belonging to the ideal generated by $P$ and $x^{...
- 109k
14
votes
Accepted
Does this degree 12 genus 1 curve have only one point over infinitely many finite fields?
The polynomial you wrote is the product of the four polynomials $x^3 - r y^3 - z^3$, where $r$ is a root of the polynomial $t^4 - t^3 + t^2 - t + 1$. I did not read your reference, but likely they ...
- 156
14
votes
What is a function field analog of Giuga's conjecture?
Here is a proof of the experimental fact Joe Silverman discovered.
Let $f$ be a polynomial of degree $d$ in $R:=\mathbb{F}_q[x]$. For $I=(f)$, the sum
$$1+\sum_{a \in R/I} a^{\# R/I - 1}$$
is equal, ...
- 14.6k
13
votes
Cubic polynomials over finite fields whose roots are quadratic residues or non-residues
Yes, it does. Here is a stronger claim:
Theorem 1. Let $F$ be a field of characteristic $\neq2$. Let $c\in F$. Let
$r\in F$ be a nonzero square, and let $n_{1},n_{2}\in F$ be such that the
...
- 33.8k
13
votes
Accepted
Cubic polynomials over finite fields whose roots are quadratic residues or non-residues
EDIT: Following a clever observation of user44191 in the comments:
If $f(x)$ is a monic polynomial, and $c$ a number, then the polynomial $xf(x)^2+c$ has a similar property to your example (the case $...
- 148k
13
votes
Accepted
Subgroups of the multiplicative group of a finite field satisfying a certain additive property
If we let $S$ be the set of characters of $\mathbb F_p^\times$ trivial on $G$ then $$\sum_{\chi \in S} \chi(g) = \begin{cases} \frac{p-1}{|G|} & g\in G \\ 0 & g\notin G \end{cases}$$
so $$\...
- 148k
13
votes
Accepted
Pointless groups
Perhaps I'm missing something, but it seems to me that $k=\mathbb{F}_2$, $G=\mathbb{G}_{m, k}$ works, where $H$ is the trivial subgroup.
- 23k
12
votes
Good source for representation of GL(n) over finite fields?
All finite dimensional complex representations of finite groups are equivalent to unitary representations, so the requirement that the representations be unitary is not really a restriction.
The 1955 ...
- 44.4k
12
votes
Accepted
Maximal order of elements in SL(n,q)
Yes. It is shown in the paper
Darafsheh, M.R., Order of elements in the groups related to the general linear group., Finite Fields Appl. 11, No. 4, 738-747 (2005). ZBL1147.20043.
(Theorem 1)
that ...
- 96.4k
12
votes
Accepted
Galois action on $p$-adic Tate module of Abelian variety over finite field semisimple?
You're just asking whether Frobenius acts semsimply on the $p$-adic Tate module.
We know from Tate's theorem that Frobenius acts semisimply on the $\ell$-adic Tate module, and hence satisfies some ...
- 148k
12
votes
Accepted
Is there an $\mathbb{R}$-valued cohomology theory for varieties over $\mathbb{F}_p$?
The answer is no. If $A$ is a simple abelian variety over $\mathbb{F}_p$, then $End(A)\otimes\mathbb{R}$ cannot act on a real vector space of dimension $2dim(A)$ if the center of the endomorphism ...
- 136
12
votes
Accepted
What is a function field analog of Giuga's conjecture?
The integers 1 to $n-1$ are the non-zero elements of $\mathbb Z/n\mathbb Z$. So one could take an ideal $I$ in a Dedekind domain $R$ and ask whether
$$
(*)\qquad
1+\sum_{\substack{a\in R/I\\ a\ne0\\}} ...
- 47.4k
12
votes
Accepted
p-adic analogue of octonions
Defining and classifying the octonion algebras (composition algebras of dimension $8$) over fields $k$, or, in more sophisticated terms, computing the Galois cohomology set $H^1(k, G_2)$, is the topic ...
- 32.4k
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