Log-Base 2

(AIME 1994) Find the positive integer n for which
[log2 1] + [log2 2] + [log2 3] +…+ [log2 n] = 1994
where [x] denotes the greatest integer less than or equal to x (for example [Π] = 3)
Solution:
Observe that:
[log2 1] = 0(20)
[log2 2] + [log2 3] = 1(21)
[log2 4] + [log2 5] + [log2 6] [...]

From Infinity to Divinity

There was a young fellow from Trinity (Cambridge University?)…..
…..Who took √∞ ….. But the number of digits….. Gave him the fidgets…..
He dropped Math and took up Divinity…
Hehehehe … ∞
This is the phrase at the bottom of title page of the book One, Two, Three…Infinity by George Gamow.
I enjoyed reading it, since it is about facts [...]

Some Proofs on Irrationality

Prove that √5 + √3 is irrational
(No.4 of Problem Set 10 in Numbers: Rational and Irrational by Ivan Niven)
Solution:
The argument made here is a proof by contradiction or reductio ad absurdum; That is, assuming that the proposition is false and then derive a contradiction from this assumption.
Note: My solutions are also parallel to Prof. Niven’s
First, [...]