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General Diophantine Equations

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

Concept Quizzes

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?


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