# Tagged Questions

93 views

### Find all positive integers satisfying $\frac{2^n+1}{n^2} =k$

Find all positive integers satisfying $$\frac{2^n+1}{n^2} =k$$ where $k$ is a integer. I can't just come up with a solution.
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### Solution to Diophantine equation with constraint.

solve the following equation over $z_x,z_y$ \begin{align} &az_x=bz_y\\ &\text{s.t. } a,b,z_x,z_y \in \mathbb{Z} \text{ and } 1 \le z_x \le N \text{ and } 1 \le z_y \le N \end{align} How ...
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### Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
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### Hard Simultaneous Diophantine Equations

Find all positive integers $a,b,c,d,e,f$ such that : $de^2=ab^2+1$ and $df^2=ac^2+1$. I tried subtracting them, it factors quite nicely. But after that, haven't a clue. I'm not sure if it's even ...
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### Find all non-trivial triplets $(a,b,c)$ such that $a+b^{d}\equiv0\pmod c$ for all $d>0$

Let $a,b,c$ be co-prime integers $>2$ . Find all non-trivial triplets $(a,b,c)$ such that $a+b^{d}\equiv0\pmod c$ for all $d>0$.
Waring's problem asks about natural numbers that can be represented as a sum of natural numbers all raised to the same power $k$. I'm wondering which natural numbers can be represented as a sum of ...