Kvant Math Problem 960

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kvantmathematicsolympiad

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Verdicts: FAIL + FAIL
Solve time: 29m02s
Source on kvant.digital

Problem

If the difference between the cubes of two consecutive natural numbers is the square of some natural number $n$, then the number $n$ can be represented as the sum of the squares of two consecutive natural numbers.

  1. Prove this statement.
  2. Here is one example of such numbers: $8^3-7^3=(2^2+3^2)^2$; give at least one more example.
  3. Prove that there are infinitely many such examples.

R. Lyness (Great Britain)

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