Show that if the difference of the cubes of two consecutive integers is the square of an integer, then this integer is the sum of the squares of two consecutive integers.

(The smallest non-trivial example is: 8^{3} − 7^{3} = 169. This is the square of an integer, namely 13, which can be expressed as 2^{2} + 3^{2}.)

We have (m + 1)^{3} − m^{3} = 3m^{2} + 3m + 1 = n^{2}, for some integers m and n.

Hence 12m^{2} + 12m + 4 = 4n^{2}, from which 3(2m + 1)^{2} = (2n − 1)(2n + 1).

Now, 2n − 1 and 2n + 1 are relatively prime.

(By Euclid's algorithm, their greatest common divisor divides their difference, namely 2. Since both are odd, their greatest common divisor must be 1.)

Therefore we must consider two possible cases, with a and b relatively prime:

- 2n − 1 = 3a
^{2}, 2n + 1 = b^{2} - 2n − 1 = a
^{2}, 2n + 1 = 3b^{2}

Taking the first case, we have b^{2} = 3a^{2} + 2.

This is impossible, as any square is congruent to 0 or 1, modulo 3.

Taking the second case, notice that a must be odd.

Setting a = 2k + 1, we have 2n − 1 = (2k + 1)^{2} = 4k^{2} + 4k + 1.

Hence 2n = 4k^{2} + 4k + 2 = 2[k^{2} + (k + 1)^{2}].

So n = k^{2} + (k + 1)^{2}.

Therefore, if the difference of the cubes of two consecutive integers is the square of an integer, then this integer is the sum of the squares of two consecutive integers.

Solutions of the equation are given by (m, k^{2} + (k + 1)^{2}) = (x_{n}, y_{n}), where (x_{0}, y_{0}) = (0, 1), (x_{1}, y_{1}) = (7, 13), and (x_{n+1}, y_{n+1}) = (14x_{n} − x_{n−1} + 6, 14y_{n} − y_{n−1}) for n 1.

Source: American Mathematical Monthly 57 (1950), 190.^{ }

The table below shows all non-negative solutions to (m + 1)^{3} − m^{3} = [k^{2} + (k + 1)^{2}]^{2}, for k 10^{9}.

m | k |
---|---|

0 | 0 |

7 | 2 |

104 | 9 |

1455 | 35 |

20272 | 132 |

282359 | 494 |

3932760 | 1845 |

54776287 | 6887 |

762935264 | 25704 |

10626317415 | 95930 |

148005508552 | 358017 |

2061450802319 | 1336139 |

28712305723920 | 4986540 |

399910829332567 | 18610022 |

5570039304932024 | 69453549 |

77580639439715775 | 259204175 |

1080558912851088832 | 967363152 |

- Pell's Equation
- Online Encyclopedia of Integer Sequences: A001921 and A001571

Source: Postal Problem Sheet. (Page since taken down.) (Originally due to R. C. Lyness.)