Intereting Posts

Lebesgue-integrable function
The pseudoness of pseudorandom number generators
Why is $\omega$ the smallest $\infty$?
Why (directly!) does every number divide 9, 99, 999, … or 10, 100, 1000, …, or their product?
Solving an integral (with substitution?)
Another pigeonhole principle question
$f(f(x)f(y))+f(x+y)=f(xy)$
Two covering spaces covering each other are equivalent?
lower bound for the prime number function
Splitting of a polynomial modulo primes of a ring of integers
Are there any quadratic subfields of $\mathbb{Q}(\sqrt{1+\sqrt{2}})$?
prove $f(z)=cz^n$ for some $c$.
Field Norm Surjective for Finite Extensions of $\mathbb{F}_{p^k}$
Steps to solve semi-infinite IBVP
Does $\sum{\frac{\sin{(nx)}}{n}}$ converge uniformly for all $x$ in $$

The equation $y^2 = x^3 + k$ for $k = (4n-1)^3 – 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 divids m, doesn’t have any answer and its proof can be obtained by using quadratic reciprocity law.

Do you know answers of this equation for two or three different values of $k$? In addition, do you know any reference about that?

- Normality of a certain order of an algebraic number field
- Decomposition of a prime number in a cyclotomic field
- Are distinct prime ideals in a ring always coprime? If not, then when are they?
- Why is $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$?
- “Prime decomposition of $\infty$”
- Proof that $\mathbb{Z}\left$ is a PID

- Decomposition of $l$ in a subfield of a cyclotomic number field of an odd prime order $l$
- Why is the restricted direct product topology on the idele group stronger than the topology induced by the adele group?
- Find All $x$ values where $f(x)$ is Perfect Square
- Intersection of a number field with a cyclotomic field
- Proof that $x^2+4xy+y^2=1$ has infinitely many integer solutions
- Solve in $\mathbb Z^3$.
- Existence of solutions to diophantine quadratic form
- is the number algebraic?
- Positive integer solutions of $a^3 + b^3 = c$
- Integers in biquadratic extensions

This is a famous class of elliptic curves, called Mordell’s equation, or sometimes Mordell-Bachet equation. See also here, or here for some discussions on MSE. For a specific example with $k=2000000$ see also here. A further reference is this article by Keith Conrad.

- How to efficiently generate a set uniformly distributed numbers that add to $n$.
- Forming Partial Fractions
- Calculate integrals $\int_0^1 {\frac{{\arcsin x}}{x}dx} $
- Help with combinatorial proof of identity: $\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$
- Find this limit without using L'Hospital's rule
- Do these series converge to logarithms?
- Counting necklace with no adjacent beads are of the same color
- Proof of Parseval's Theorem for Fourier Series
- Is there a short proof for the Intermediate Value Theorem
- Let $f : D \rightarrow D$ be a continuous map whose restriction to $S^1$ is the identity map. Show that $f$ must be surjective.
- for any ring $A$ the matrix ring $M_n(A)$ is simple if and only if $A$ is simple
- How to find $\lim _{ n\to \infty } \frac { ({ n!) }^{ 1\over n } }{ n } $?
- What is the periodicity of the function $\sin(ax) \cos(bx)$ where $a$ and $b$ are rationals?
- Derive an algorithm for computing the number of restricted passwords for the general case?
- How to prove that $\mathrm{Fibonacci}(n) \leq n!$, for $n\geq 0$