All Questions
11 questions
1
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0
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71
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Bias of $a^k / q$ modulo $q$?
Let $q$ be a prime. Let $0< a < q$ be an integer so that it is primitive modulo $q$. Let $k$ be a random integer up to $q-1$. Consider
$$a^k = b_k + q * c_k$$
as $k$ varies modulo $q^2$. So $b_k$...
- 591
4
votes
0
answers
175
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Intrinsic maps between complex integers modulo $p$ and integers modulo $p+2$
$\DeclareMathOperator\GF{GF}$Let $p$ and $p+2$ be twin primes. Let's assume that $-1$ is not a quadratic residue modulo $p$ (and therefore is a Q.R. modulo $p+2$).
Consider the complex numbers $a+bi$ ...
- 63.9k
0
votes
1
answer
195
views
Trying to solve for the remainder of $a^q$ modulo $q$
Let $q$ be a prime and $a$ be a number from $0$ to $q-1$ (not an equivalence class).
The elements $a^q$ are exactly the elements of order $q-1$ modulo $q^2$.
I'm trying to solve the equation:
$$a+2*\...
- 34.3k
3
votes
0
answers
73
views
Is the discrete logarithm equivalent to solving polynomial discrete logarithms?
Suppose we can quickly solve the discrete logarithm modulo $p$. Let's say $2$ is a generator so we can quickly find $l$ for which $2^l =h$ for any given target $h$.
An interesting observation is that ...
- 591
2
votes
0
answers
145
views
What is the periodicity of $((a^n \text{ modulo } p) \text{ modulo } q)$
This feels like it should be elementary but it came up in my research and I was not able to solve it.
We can ask this question for any $p$ and $q$ but,let $p$ and $q$ be primes for simplicity. The ...
- 591
3
votes
0
answers
269
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Finding (and saturating) a sharp Babenko-Beckner inequality for finite fields
My question is a follow-up to Abdelmalek Abdesselam's recent post
What makes Gaussian distributions special? Local field version?
asking about various characterizations of (real-valued) Gaussian ...
- 2,137
2
votes
0
answers
243
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Sums of squares in fields
Which fields $k$ have the property that any sum of squares is a square ?
Are there elegant characterizations and/or classifications known for this type of field ?
And what if we replace "fields" by "...
- 4,547
2
votes
0
answers
228
views
Satake correspondence for groups over finite field
I asked the same question in MSE, but I didn't get any answer. So I decided to post it here, too.
In Langlands' program, Satake correspondence gives a correspondence between unramified ...
- 2,215
1
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0
answers
69
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On equality of two quotients of a congruence subgroup
Related question: Non-torsion part of the abelianisation of congruence subgroups
Let $A = \mathbb{F}_q[T]$ be the ring of polynomials with coefficients in a finite field, with $N$ a nonconstant ...
- 63.9k
5
votes
2
answers
571
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Exceptional isomorphisms between finite simple Chevalley groups
Steinberg's "Lectures on Chevalley Groups"
https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf
contain ``a complete list of isomorphisms" among the various finite simple Chevalley ...
- 15.5k
20
votes
2
answers
2k
views
Sums of powers mod p
For prime $p > 7$ with $p-1=rs$, $r>1$, $s>1$, let $A=\{x^r|x \in \mathbb{Z}_p\}$ and $B = \{x^s|x \in \mathbb{Z}_p\}$. If $g$ is a primitive root mod $p$ then $A = \{0\} \cup \{g^{ir}|0 \leq ...
- 3,320