×

## General Diophantine Equations

We have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.

# Level 3

1729 can be expressed as the sum of two perfect cubes in two distinct ways.

How many such numbers exist?

Hint: Given one set of such numbers, how could you construct another set?

If $$a$$ and $$b$$ are integers such that $$a^3+b^3=2015$$, then what is $$a+b$$?

$\LARGE{2 + 2 = 2 \times 2 \\ 1 + 2 + 3 = 1 \times 2 \times 3}$

These are sets of two and three positive integers whose sum equals their product.

Find a set of four positive integers whose sum equals their product. What is their sum?

Find the positive integer $$n$$ such that $$n^3 + 2n^2 + 9n + 8$$ is a perfect cube.

$a + b + c + d + e = a \times b \times c \times d \times e$

How many unordered 5-tuples of positive integers are there which satisfy the above equation?

×