show/hide this revision's text 3 Item #3 added.

Here are a couple of few others:

  1. Let $H_n=\sum_{j=1}^n 1/j$. Then for all $n\geq 1$, $$ \sum_{d|n}d\leq H_n+(\log H_n)e^{H_n}. $$ Jeff Lagarias showed that this is equivalent to the Riemann hypothesis!

  2. Let $x_0=2$, $x_{n+1}=x_n-\frac{1}{x_n}$ for $n\geq 0$. Then $x_n$ is unbounded.

  3. The largest integer that cannot be written in the form $xy+xz+yz$, where $x,y,z$ are positive integers, is 462. It is known that there exists at most one such integer $n>462$, which must be greater than $2\cdot 10^{11}$. See J. Borwein and K.-K. S. Choi, On the representations of $xy+yz+xz$, Experiment. Math. 9 (2000), 153-158; http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.em/1046889597.
show/hide this revision's text 2 $n\geq 1$ replaced with $n\geq 0$

Here are a couple of others:

  1. Let $H_n=\sum_{j=1}^n 1/j$. Then for all $n\geq 1$, $$ \sum_{d|n}d\leq H_n+(\log H_n)e^{H_n}. $$ Jeff Lagarias showed that this is equivalent to the Riemann hypothesis!

  2. Let $x_0=2$, $x_{n+1}=x_n-\frac{1}{x_n}$ for $n\geq 1$0$. Then $x_n$ is unbounded.

show/hide this revision's text 1 [made Community Wiki]

Here are a couple of others:

  1. Let $H_n=\sum_{j=1}^n 1/j$. Then for all $n\geq 1$, $$ \sum_{d|n}d\leq H_n+(\log H_n)e^{H_n}. $$ Jeff Lagarias showed that this is equivalent to the Riemann hypothesis!

  2. Let $x_0=2$, $x_{n+1}=x_n-\frac{1}{x_n}$ for $n\geq 1$. Then $x_n$ is unbounded.