Intereting Posts

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I ask this question based on a comment of David Speyer in another question. What primes are of the form

$$

\frac{p^2-1}{q^2-1}

$$

where $p$ and $q$ are prime?

The first prime not apparently of this form is 17. The Diophantine equation

$$

p^2-17q^2+16=0

$$

has solutions following a linear recurrence relation which has no primes in the first 1000 terms (only $(\pm1, 1)$ seeds may contain primes). But perhaps there is a better way to go about this?

- $x^y = y^x$ for integers $x$ and $y$
- $|2^x-3^y|=1$ has only three natural pairs as solutions
- How can $p^{q+1}+q^{p+1}$ be a perfect square?
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- Integers can be expressed as $a^3+b^3+c^3-3abc$
- solving cubic diophantine equation

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- Convergence or divergence of $\sum\limits_n(-1)^{\pi(n)}\frac1n$ where $\pi(n)$ is the number of primes less than or equal to $n$
- Is $e^{n\pi}$ transcendental?
- Conjecture involving semi-prime numbers of the form $2^{x}-1$
- Norm of ideals in quadratic number fields
- The n-th prime is less than $n^2$?
- How many solutions are there to $F(n,m)=n^2+nm+m^2 = Q$?
- What is the highest power of 2 dividing 100!
- How prove this $\prod_{1\le i<j\le n}\frac{a_{j}-a_{i}}{j-i}$ is integer

I think for the equation:

$$\frac{x^2-1}{y^2-1}=k$$

It is necessary to record decisions. We will use the solutions of the Pell equation.

$$p^2-ks^2=1$$

And then the solutions are of the form:

$$x=-p^2+2kps-ks^2$$

$$y=p^2-2ps+ks^2$$

- Find the values of $p$ such that $\left( \frac{7}{p} \right )= 1$ (Legendre Symbol)
- Infinitely times differentiable function
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- Evaluating an integral across contours: $\int_C\text{Re}\;z\,dz\,\text{ from }-4\text{ to } 4$
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- Using $\epsilon$-$\delta$ definition to prove that $\lim_{x\to-2}\frac{x-1}{x+1}=3$.
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