Number Theory

General Diophantine Equations

General Diophantine Equations: Level 3 Challenges

         

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 aa and bb are integers such that a3+b3=2015a^3+b^3=2015, then what is a+ba+b?

2+2=2×21+2+3=1×2×3\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 nn such that n3+2n2+9n+8n^3 + 2n^2 + 9n + 8 is a perfect cube.

a+b+c+d+e=a×b×c×d×e 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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