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Results tagged with nt.number-theory
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user 48142
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
7
votes
Accepted
Growth order of numbers whose prime factors are all congruent to +1 or -1 modulo 8
This is an old question (and according to this MO question, the result you seek was proven by Landau). In particular, it follows from this that if $S$ is a set of arithmetic progressions containing a …
- 19.9k
38
votes
Accepted
When does a Catalan number equal a Fibonacci number?
A result of the type you seek follows easily from Carmichael's theorem, that if $m > 12$, then there is a prime $p$ that divides $F_{m}$, but does not divide $F_{k}$ for $k < m$.
Suppose $C_{n} = \b …
- 19.9k
5
votes
Accepted
Why is this function a modular function of level $5$?
Here's a fairly straightforward way to show that $\phi$ is modular of level $5$ using Siegel functions.
Claim: The function $f(\tau)$ is a modular function for $\Gamma(5)$ if and only if $f(5\tau)$ is …
- 19.9k
21
votes
Is there a real nonintegral number $x >1$ such that $\lfloor x^n \rfloor$ is a square intege...
At the request of the OP, I am turning my comment into an answer. It is possible to have $\lfloor x^{n} \rfloor$ close to a square for all positive integers $n$. For example, if $x = \frac{7 + 3 \sqrt …
- 19.9k
2
votes
Accepted
Imaginary quadratic fields with $\ell$-indivisible class number
Here's an elementary argument. For $\ell < 41$, $K = \mathbb{Q}(\sqrt{-163})$ works. For $\ell = 41$, $K = \mathbb{Q}(\sqrt{-3})$ works. Assume then that $\ell \geq 43$.
Choose an integer $1 \leq n \l …
- 19.9k
8
votes
Accepted
An old conjecture of M.Newman
First, I think your definition of $H_{n}$ does not agree with Newman's definition. Newman says the following: "Let $H_{n} \subset G_{n}$ be the set of functions of $G_{n}$ with non-negative valence at …
- 19.9k
32
votes
Only odd primes?
If $k$ is odd and not a perfect square, then the sets are disjoint. In particular, if $\alpha = \frac{k - \sqrt{k}}{\frac{k-1}{2}}$ and $\beta = \frac{k + \sqrt{k}}{\frac{k-1}{2}}$, then $\alpha$ and …
- 19.9k
7
votes
Can the generalized divisor summatory function $D_z$ be expressed explicitly in terms of Zet...
The short answer is that, in all likelihood, a formula of the type you seek only exists if $z$ is a negative integer.
The way to approach a question like this is to first note that $\sum_{n=1}^{\inf …
- 19.9k
2
votes
Accepted
Small Galois group solution to Fermat quintic
I have answers to your first two questions, and some insight into the third.
First, there is a quartic in the family above with Galois group contained in $A_{4}$. One example is found by taking $q = 1 …
- 19.9k
16
votes
Accepted
Prove that $1$ is the sum of three tetrahedral numbers infinitely many different ways
There are infinitely many solutions. I'll show below that there are infinitely many positive integers $k$ for which $93k^{2} - 288k + 276 = z^{2}$ for some positive integer $z$. From such a $z$, we ge …
- 19.9k
8
votes
Examples of models for modular curves
Here's an example. Let's take $\Gamma = \Gamma_{0}(4)$, and
$\Gamma' = \Gamma(2)$. We'll let $\alpha = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$, so $\Gamma' = \alpha \Gamma \alpha^{-1}$. The func …
- 19.9k
8
votes
Accepted
Equivalent notions of congruence for elliptic curves over $\mathbb{Q}$
I found myself wondering the same thing a couple of weeks ago. Even with the restriction that $p > 2$ and the residual representation is irreducible, it does not follow that $E_{1}$ and $E_{2}$ have t …
- 19.9k
16
votes
Accepted
$x^3+x^2y^2+y^3=7$, and solvable families of Diophantine equations
(a) No. There are no integer solutions. The curve $C$ you give has genus $3$ and it has an obvious automorphism $\phi(x,y) = (y,x)$. The quotient curve is an elliptic curve. In particular, if you let …
- 19.9k
1
vote
Bounds on largest possible square in sum of two squares
Rather than discuss $\max b_{i}$, I'll discuss the equivalent question of bounding $\min a_{i}$. The ABC conjecture implies that for all $\epsilon > 0$, $\min a_{i} \gg (c^{2}+1)^{n/2 - 1 - \epsilon}$ …
- 19.9k
26
votes
Accepted
Does the set of square numbers adhere to Benford's law in every base?
No. Benford's law works well for sequences that grow exponentially, and the squares grow too slowly.
In particular, fix a base $b \geq 3$, consider the case of $d = 1$, and choose $n = 2 \cdot b^{2k}$ …
- 19.9k