I’m working on a paper in which I end up considering the biquadratic rational curve $$u^2v^2 - u^2 - v^2 - 6uv + 8 = 0. \tag{$1$}$$ To complete the remainder of my proof/method, I need to prove that it has rank 1. I believe it can be transformed to the Weierstrass form $$Y^2=X^3-32X,$$ and then...

The above given equation solution is very easy just make it to an elliptic curve. For $$ W^{4}+X^{4}=Y^{2}+Z^{4} $$ Divide both sides $Z^{4}$, we wil get $$ \left(\frac {W}{Z}\right)^4 + \left(\frac{X}{Z}\right)^4 = \left(\frac{Y}{Z^2}\right)^2 +1 $$ If we substitute $\frac{W}{Z} = (u+v)$,...

Problem:I am looking for help with the following Diophantine equation: $$y^2 = x^3 - x^2 + 16$$ By working through the equation, I have successfully found 8 distinct non- negative integer solutions. The largest value of $x$ among all the solutions I found is $x = 112$ (which gives $y = 1180$)....

Let $a$ be a positive integer, not a cube, so that $\alpha=\sqrt[3]a$ is irrational, and write $$R={\mathbb Z}[\alpha]=\{\,x+y\alpha+z\alpha^2\ |\ x,y,z\in\mathbb{Z}\,\}\ .$$ Let $\beta=x+y\alpha+z\alpha^2$ be a unit in $R$ with norm (product of conjugates) equal to $1$ (and not $-1$). Then...

Tate's $p$-adic uniformisation theorem for elliptic curves goes as follows: Let $K$ be a $p$-adic field, let $E/K$ be an elliptic curve with $v_K(j) \ge 0$, and let $\gamma(E/K)=-c_4/c_6 \in K^{\times}/(K^{\times})^2$. a) There is a unique $q \in K^{\times}$ with $|q|<1$ such that $E$ is...

Definition: Let $p_i$ be the $i$-th prime, $\pi(p_i)=i$ the prime counting function, and $g_i=p_{i+1}-p_i$ the prime gap. Observation - "Beta Density Law": For all primes $p_i$ with $i \leq 168$, i.e. $p_i \leq 997$, the gap satisfies: $$g_i \approx 0.065 \cdot \frac{p_i}{\pi(p_i)}$$ Note on...

I am studying a family of non‑homogeneous linear complex differential equations and encountered the following limit. I would like an explicit counterexample, if one exists. We consider $\eta \in L^\infty([1,+\infty))$ satisfying the following hypothesis $(H)$: $$ \exists \rho_\eta \in...

In my research project I am looking at a lower bound for Kloosterman sums, which are sums of roots of unity. The best known lower bound for a sum of $k$ $N$th roots of unity is $k^{-N}$, which comes from a simple algebraic number theory argument. In a 1986 paper, "How Small Can a Sum of Roots of...