Without brute force, how can we show that 1729 is the smallest number that can be expressesed as the sum of two cubes in two different ways?
103150
Proof that $1729$ is the smallest taxicab number
84491
Finding the smallest positive integer $n$ such that $S_n$ contains an element of order 60.
6366
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
127701
How many distinct ways are there to $2$-color the $8$ vertices of a cube?
2786
The staircase paradox, or why $\pi\ne4$
22973
Show that an integer of the form $8k + 7$ cannot be written as the sum of three squares.
146241
If $a \in \Bbb Z$ is the sum of two squares then $a$ can't be written in which of the following forms?
140433
The number of ways of writing an integer as a sum of two squares
112644
The product of two numbers that can be written as the sum of two squares
18402
Sum of the reciprocals of divisors of a perfect number is $2$?
94923
Using the Pigeonhole Principle to show that $2$ of any $n+1$ numbers from $\{1,2,\ldots,2n\}$ sum to $2n+1$
114863
Representing a given number as the sum of two squares.
143113
Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$
101832
Proof that sum of first $n$ cubes is always a perfect square
33380
Can RUBIK's cube be solved using group theory?
64803
Counting two ways, $\sum \binom{n}{k} \binom{m}{n-k} = \binom{n+m}{n}$
98082
Another combinatorics problem: $\sum\limits_{k = 0}^n (-1)^k \binom{2n-k}k2^{2n-2k}=2n+1$