Intereting Posts

Existence of ultrafilters
What is the infinite sum
How to show that every $\alpha$-Hölder function, with $\alpha>1$, is constant?
Is Inner product continuous when one arg is fixed?
Practical system with the following ODE form
IMO 1988 question No. 6 Possible values of $a$ and $b$, $\displaystyle\frac{a^2+b^2}{ab+1}$
Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$
“sheaf” au sens de Serre
Improper integral $\int_0^\infty \frac{\sin(x)}{x}dx$ – Showing convergence.
How to efficiently generate a set uniformly distributed numbers that add to $n$.
How can I tell if $x^5 – (x^4 + x^3 + x^2 + x^1 + 1)$ is/is not part of the solvable group of polynomials?
Finding a nonempty subset that is not a subgroup
Can anybody explain about real linear space and complex linear space?
Normality of a certain order of an algebraic number field
Proof of Frullani's theorem

Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive integer values of $t$ the respective specialization has a point over $K$?

(I expect the answer to be “such a $K$ is not known” and the problem of similar difficulty as that for $K = \mathbb{Q}$ but have not seen an expert comment on it so far.)

Thank you.

- Algebraic Curves and Second Order Differential Equations
- Set that is not algebraic
- Nonsingular projective variety of degree $d$
- Proving that the genus of a nonsingular plane curve is $\frac{(d-1)(d-2)}{2}$
- When do equations represent the same curve?
- Why is a genus 1 curve smooth and is it still true for a non-zero genus one in general?

- Valuation rings and total order
- what is genus of complete intersection for: $F_1 = x_0 x_3 - x_1 x_2 , F_2 = x_0^2 + x_1^2 + x_2^2 + x_3^2$
- What is an intuitive meaning of genus?
- Why is a genus 1 curve smooth and is it still true for a non-zero genus one in general?
- Rational map on smooth projective curve
- projective cubic
- Calculating the projective closure with more than one generator
- How to Compute Genus
- Why is Klein's quartic curve not hyperelliptic
- In what senses are archimedean places infinite?

Assuming the parity conjecture (known to be a consequence of the

BSD

conjecture)

for

“congruent

number curves”

$E_d: d \cdot Y^2 = X^3 – X$,

you can even take $K = {\bf Q}$ and

$$

F(x,y) = 4x^2 + 1 + (8t-1) x y^2.

$$

The curve $F(x,y) = 0$ is birational with the elliptic curve

$(8t-1) Y^2 = X^3 + 4X$ (let $x=1/X$ and $y=Y/X$),

which in turn is 2-isogenous with $E_{8t-1}: (8t-1) Y^2 = X^3 – X$.

The only ${\bf Q}(t)$-rational points of the curve $(8t-1) Y^2 = X^3 + 4X$

are the torsion points at $(0,0)$ and infinity, and I chose coordinates

$X,Y$ that put both of these points on the line at infinity, so there are

no finite solutions of $F(x,y) = 0$. On the other hand, if $t$ is

a positive integer then $|8t-1| \equiv 7 \bmod 8$, so $E_{8t-1}$

has sign $-1$, whence under BSD it has odd (and thus positive) rank

over $\bf Q$, so infinitely many rational points.

[This example has no exceptional $t$, but it’s clear *a priori*

that if there’s an example with only finitely many exceptional $t$

then one can translate $t$ to get a “new” $F$ that has none.]

The complexity arises because not solve such equations. Usually use of modular arithmetic. Although quite often it is not able to give the formula of the solution.

We give some examples.

Unique Integer solution of a non-linear equation – sometimes it is possible to write a simple formula.

How to find $a,b\in\mathbb{N}$ such that $c = \frac{(a+b)(a+b+1)}{2} + b$ for a given $c\in\mathbb{N}$ – or like this.

Diophantine equation: $(x-y)^2=x+y$ – or like this.

Perfect Square relationship with no solutions – for some special cases you can get the formula.

How to solve an equation of the form $ax^2 – by^2 + cx – dy + e =0$? – for some special cases you can get the formula.

How can I solve equation $x^2 – y^2 -2xy – x + y = 0$? – you can consider this simple.

Proving a Pellian connection in the divisibility condition $(a^2+b^2+1) \mid 2(2ab+1)$ – you can consider this simple.

Existence of $x,y$ Satisfying Diophantine Equation – you can consider this simple.

The system of genus characters determined by a binary quadratic form – there are written the same formula.

I think enough. There’s formula for binary quadratic forms. Usually only for this form can write formulas of solutions.

- Function whose third derivative is itself.
- The value of a limit of a power series: $\lim\limits_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$
- Polynomials such that roots=coefficients
- Fermat's little theorem proof by Euler
- How to prove combinatorial identity $\sum_{k=0}^s{s\choose k}{m\choose k}{k\choose m-s}={2s\choose s}{s\choose m-s}$?
- Weighing Pool Balls where the number of balls is odd
- Prove that $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$
- formula for summation notation involving variable powers
- Rational + irrational = always irrational?
- How many numbers are in the Fibonacci sequence
- “Eigenrotations” of a matrix
- An analogue of Hensel's lifting for Fibonacci numbers
- free subgroups of $SL(2,\mathbb{R})$
- (From Lang $SL_2$) Orthonormal bases for $L^2 (X \times Y)$
- Showing a diffeomorphism extends to the neighborhood of a submanifold