Sums of two cubes

I will connect the following questions and answers.

1. 42 = (12602123297335631)^3 + (80435758145817515)^3 + (-80538738812075974)^3.

2. https://people.math.harvard.edu/~alpoge/fun/fruit%20for%20thought.jpeg , aka:
Are there positive integers x, y, z such that:
x / (y + z) + y / (x + z) + z / (x + y) = 4?

3. The Birch and Swinnerton-Dyer conjecture.

4. Hilbert’s tenth problem, aka:
Is there a computer program which “solves all Diophantine equations”?

5. 0% of integers are a sum of two squares (integral or rational).

6. A positive proportion of integers are a sum of two rational cubes.

For more information, please see: https://people.math.harvard.edu/~ana/ons/