https://www.youtube.com/watch?v=O4UpNSlzKAM

Let \(a_n\) denote the nearest integer to \(\log_{168}\left(927^n\right)\), and let \(b_n=a_{n+1}-a_n\). The first few terms in the sequence \(\{b_n\}\) are \(\{b_n\} = 2,1,1,2,1,1,2,1,1,2,1,1,\ldots \)

which appears to be a sequence of period 3. Does the repeating 2,1,1 pattern continue indefinitely? Prove your answer.

I discovered a fun identity the other day. Probably every Russian schoolchild knows this, already but I didn't: For every positive integer m,

\(\sum_{k=1}^m \lfloor m/k\rfloor = \sum_{k=1}^m \sigma_0(k)\),

where \(\sigma_0(k)\) is the number of positive divisors of k.

I was working on something that led me to realize this must be true, but if I were given the identity cold and asked to prove it I'm not sure how well I'd have fared!

https://en.wikipedia.org/wiki/Group_of_Lie_type#Notation_issues

https://www.amazon.com/Red-Ben-Fox-Oak-Ridge/dp/1372054154

Cantor's diagonal argument for non-denumerability boils down to "Every function from the natural numbers to the reals fails to be surjective."

Is there an equally lovely but underappreciated argument that does "Every function from the real numbers to the natural numbers fails to be injective?" If not, why is that direction more difficult?

- Academic Homepage
- http://homepages.gac.edu/~jsiehler/

Herzlich Willkommen in Minnesota!

Joined Jan 2019