Number Theory
# General Diophantine Equations

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 \(a+b\)?

\[\LARGE{2 + 2 = 2 \times 2 \\ 1 + 2 + 3 = 1 \times 2 \times 3}\]

In the above equations, there are respectively 2 and 3 positive integers whose sum is equal to their product.

Find 4 positive integers whose sum is equal to their product. Enter the answer as 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?

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