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Results tagged with number-theory
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user 298680
Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.
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Primes such that $a_p=0$ for a particular elliptic curve
Some context
It can be proved that for the elliptic curve given by $E :y^2 = x^3-x$, the number of points of $E$ in $\Bbb P^2(\Bbb F_p)$, denoted by $|E(\Bbb F_p)|$, is $p+1$ if and only if $p \equiv …
- 23.9k
1
vote
Accepted
Is the set of real algebraic numbers in $(0,1)$ the same as the set of fractional parts of r...
Let $A$ the set of algebraic numbers between $0$ and $1$ and let
$$B = \{\{x\} \mid x>1 \text{ algebraic }\}$$
where $\{x\}$ denotes the fractional part of $x$.
Then $A=B$.
$\subset$ : if $x \in A$ …
- 23.9k
6
votes
Accepted
Cardinal Numbers and powers related to them
First of all, if $a$ and $b$ are two cardinal numbers, then we define $a^b$ to be the cardinality of the set of all functions from $b$ to $a$. In particular, you can show that $2^A$ is the cardinality …
- 23.9k
5
votes
Why does the coefficient field of a normalized Hecke form a number field?
Consider the integral Hecke algebra $T = \Bbb Z[T_n : n \geq 1]$, which is a commutative $\Bbb Z$-subalgebra of
$A := \mathrm{End}_{\Bbb C\text{-lin.}}\left[ S_k(\Gamma) \right]$, where $\Gamma := \G …
- 23.9k
7
votes
What is the characteristic of p-adic number fields?
The ring $\Bbb Z_p$ is not a field, since $p$ is not invertible (its valuation is $1$, while the valuation of any unit is $0$). The fraction field is $\Bbb Q_p$, it is an infinite extension of $\Bbb Q …
- 23.9k
2
votes
Accepted
Real numbers of the form: "difference of two linearly independent transcendental numbers"
I will follow the idea of this answer by barak manos.
If $x \neq 0$ is algebraic, you can take the transcendental numbers $a=x+\pi$ and $b=-\pi$, which are $\Bbb Q$-linearly independent.
Indeed, if …
- 23.9k
1
vote
Accepted
Next step of the two recent questions on differences of transcendental numbers
If $x \neq 0$ is algebraic, then $x=x+\pi \;-\; \pi$ and $\dfrac{x+\pi}{\pi} = \dfrac{x}{\pi}+1$ is transcendental, otherwise $\pi$ would be algebraic.
Suppose that $x$ is transcendental. Since the t …
- 23.9k
2
votes
$a^{n-1}=1\pmod n$ but $a^d\neq 1 \pmod n$ for every proper divisor $d$ of $n-1$. Prove that...
Here are some hints:
— What is the order of $[a]_n$ in $(\Bbb Z/n\Bbb Z)^{\times}$ ? Recall that in any group $G$, $g^k=1$ iff the order of $g$ divides $k$.
— The order of $[a]_n$ divides the order …
- 23.9k
3
votes
Accepted
Cancellation in quotient of fractional ideals
Here is the main statement:
Proposition: Let $R$ be a Dedekind domain (for instance the ring of integers $R=\mathcal O_K$ of some finite extension $K$ of $\Bbb Q$).
Let $\mathfrak{a}$, $\mathfra …
- 23.9k
8
votes
Accepted
Alternative form of Eisenstein integers
Let $p,q$ be integers. Then, as you tried, we have
$$p+q\omega=
p+\dfrac{-q+qi\sqrt 3}{2}=
\dfrac{2p-q+qi\sqrt 3}{2}$$
Let $a=2p-q$ and $b=q$. What can you conclude from here?
- 23.9k
6
votes
Accepted
Intuition behind supersingular reduction of elliptic curves
Here is the typical heuristic reasoning. Let $E / \Bbb Q$ be an elliptic curve.
The starting point is that for any prime $p$ of good reduction, Hasse bound tells us that $|a_p| \leq 2 \sqrt p$ where $ …
- 23.9k
7
votes
Accepted
Prove linear combinations of logarithms of primes over $\mathbb{Q}$ is independent
If $\sum_{j=1}^{t}x_j\log(p_j)=0$
then
$\sum_{j=1}^{t}y_j\log(p_j)=0$ where $y_j \in \Bbb Z$ is the product of $x_j$ by the common denominator of the $x_j$'s.
Therefore $\log\left(\prod_{j=1}^t p_j^ …
- 23.9k
25
votes
Accepted
Do there exist numbers normal in every base except for one?
This is not possible. In Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences (1974), you can find the following exercise, on page $77$:
If $b_1$ and $b_2$ are integers $≥2$, such th …
- 23.9k
3
votes
Is the last digit of this number :$ {{4^4}^n}+1 $ always $7 $ for $n>1$ and could this be pr...
Yes, it is true. For $n≥1$, $4^{(4^n)} = 16^{2^{2n-1}}$ is a power of 16.
Since $16^2 = 256 \equiv 6 \pmod {10}$ and $16\cdot 6 = 96 \equiv 6 \pmod{10}$, every power of $16$ ends with $6$. Therefore …
- 23.9k
3
votes
Accepted
Relation between decomposition field and complete decomposition of a prime ideal
Since $G_{\mathfrak p}$ is the stabilizer of $\mathfrak p$, we know that $G/G_{\mathfrak p}$ is in bijection with the orbit of $\mathfrak p$, so $[G:G_{\mathfrak p}]$ is the number $r$ of primes above …
- 23.9k