More generally we can ask, for an n-set of distinct integers, how many of the sums of k of the n elements can be square? For example, for the set { -9, 3, 6, 7 } the sum of any 3 elements is square. We can also generalize by wanting the sums to be cubes, triangular numbers, primes, or having some other distinguishing property.

n | number of sums | smallest such set | Author |
---|---|---|---|

2 | 1 | { -1, 1 } | Leo Moser |

3 | 3 | { -4, 4, 5 } | Berend van der Zwaag |

4 | 6 | { -94, 95, 130, 194 } | Leo Moser |

5 | 10 | { -4878, 4978, 6903, 12978, 31122 } | Jean-Louis Nicolas |

6 | 15 | { -15863902, 17798783, 21126338, 49064546, 82221218, 447422978 } | Jean Lagrange |

n | number of sums | smallest such set | Author |
---|---|---|---|

2 | 1 | { -1, 0 } | Bryce Herdt |

3 | 3 | { -2, 1, 2 } | Erich Friedman |

4 | 6 | { -32, 32, 67, 256 } | Berend van der Zwaag |

n | number of sums | smallest such set | Author |
---|---|---|---|

3 | 1 | { -1, 0, 1 } | Bryce Herdt |

4 | 4 | { -9, 3, 6, 7 } | Erich Friedman |

Richard Sabey sent an analysis of sums of n-1 numbers from an n-set adding to n consecutive squares. This can be accomplished for n=1, 2, 3, 4, 6, 8, 9, 11, 12, 14, 15, 18, 20, É

n | number of sums | smallest such set | Author |
---|---|---|---|

2 | 1 | { -1, 1 } | Erich Friedman |

3 | 3 | { -1, 1, 2 } | Bryce Herdt |

4 | 6 | { -9, 12, 24, 54 } | Bryce Herdt |

n | number of sums | smallest such set | Author |
---|---|---|---|

2 | 1 | { -1, 0 } | Bryce Herdt |

3 | 3 | { -3, 2, 3 } | Erich Friedman |

4 | 6 | { -14, 16, 28, 49 } | Erich Friedman |

n | number of sums | smallest such set | Author |
---|---|---|---|

3 | 1 | { -1, 0, 1 } | Bryce Herdt |

4 | 4 | { -7, 2, 5, 8 } | Bryce Herdt |

n | number of sums | smallest such set | Author |
---|---|---|---|

2 | 1 | { 0, 2 } | Bryce Herdt |

3 | 3 | { 0, 2, 3 } | Bryce Herdt |

4 | 5 | { 0, 2, 3, 5 } | Berend van der Zwaag |

5 | 7 | { -4, -2, 4, 7, 9 } | Berend van der Zwaag |

6 | 10 | { -3, -1, 3, 5, 8, 14 } | Berend van der Zwaag |

7 | 14 | {-6, -4, 6, 8, 11, 23, 35} | Berend van der Zwaag |

8 | 18 | {-14, -8, 10, 16, 21, 27, 31, 37} | Berend van der Zwaag |

9 | 22 | {-21, -17, -11, 13, 19, 24, 28, 34, 40} | Berend van der Zwaag |

10 | 27 | {-30, -24, -4, 6, 26, 32, 35, 41, 47, 77} | Berend van der Zwaag |

n | number of sums | smallest such set | Author |
---|---|---|---|

2 | 1 | { 0, 1 } | Erich Friedman |

3 | 3 | { 0, 2, 4 } | Erich Friedman |

4 | 6 | { -2, 4, 6, 12 } | Erich Friedman |

5 | 10 | { -2, 4, 6, 12, 24 } | Berend van der Zwaag |

6 | 15 | { -2, 4, 8, 14, 32, 38 } | Berend van der Zwaag |

7 | 21 | { -2, 4, 12, 18, 24, 48, 54 } | Berend van der Zwaag |

8 | 28 | { -2, 4, 8, 14, 32, 38, 74, 98 } | Berend van der Zwaag |

9 | 36 | { -2, 4, 8, 14, 32, 38, 74, 98, 158 } | Berend van der Zwaag |

10 | 45 | { -1, 11, 29, 59, 71, 167, 211, 389, 431, 449 } | Berend van der Zwaag |

n | number of sums | smallest such set | Author |
---|---|---|---|

3 | 1 | { -1, 1, 2 } | Erich Friedman |

4 | 4 | { -4, 2, 4, 5 } | Berend van der Zwaag |

5 | 10 | { -9, 3, 9, 11, 17 } | Erich Friedman |

6 | 19 | { -5, 1, 7, 15, 21, 51 } | Erich Friedman |

7 | 31 | {-41, 19, 25, 29, 35, 53, 119} | Berend van der Zwaag |

8 | 49 | {-111, 27, 57, 87, 97, 127, 265, 295} | Berend van der Zwaag |

n | number of sums | smallest such set | Author |
---|---|---|---|

2 | 1 | { -1, 1 } | Erich Friedman |

3 | 3 | { -13, 13, 14 } | Erich Friedman |

4 | 6 | { -35780, 4693243, 11888132, 70993724 } | Bryce Herdt |

n | number of sums | smallest such set | Author |
---|---|---|---|

2 | 1 | { -1, 0 } | Bryce Herdt |

3 | 3 | { -4, 3, 4 } | Erich Friedman |

4 | 6 | { -27224065, 31881527, 39076416, 43805439 } | Bryce Herdt |

n | number of sums | smallest such set | Author |
---|---|---|---|

3 | 1 | { -1, 0, 1 } | Bryce Herdt |

4 | 4 | { -15, 4, 11, 12 } | Erich Friedman |

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 10/13/10.