Challenge Level

Ben Twigger and Tom Ruffett from Ousedale School, Milton Keynes set up a spreadsheet which shows that

$$1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n = 2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$$

for values of $n$ from $n = 1$ to $n = 6$ but the two expressions are not equal for $n = 7$ or $n = 8$. Ben and Tom found this hard to believe themselves even though they had the evidence from their own work. The two columns headed 'sum' give the totals of the expressions on the left hand side and the right hand side for each value of $n$. The column headed 'Difference' gives the differences between these two totals for each value of $n$. Notice that the difference is 0 for $n = 1, 2, 3, 4, 5, 6$ showing that the expressions are equal for these values of $n$ and the difference is not zero for $n=7$ or $n=8$ showing that these expressions are not equal for these values of $n$.

n | 1 | 19 | 20 | 51 | 57 | 80 | 82 | sum | Difference |
---|---|---|---|---|---|---|---|---|---|

1 | 1 | 19 | 20 | 51 | 57 | 80 | 82 | 310 | 0 |

2 | 1 | 361 | 400 | 2601 | 3249 | 6400 | 6724 | 19736 | 0 |

3 | 1 | 6859 | 8000 | 132651 | 185193 | 512000 | 551368 | 1396072 | 0 |

4 | 1 | 130321 | 160000 | 6765201 | 10556001 | 40960000 | 45212176 | 103783700 | 0 |

5 | 1 | 2476099 | 3200000 | 345025251 | 601692057 | 3276800000 | 3707398432 | 7936591840 | 0 |

6 | 1 | 47045881 | 64000000 | 17596287801 | 34296447249 | 2.62E+11 | 3.04E+11 | 6.18E+11 | 0 |

7 | 1 | 893871739 | 1280000000 | 8.97E+11 | 1.95E+12 | 2.10E+13 | 2.49E+13 | 4.88E+13 | 36021585600 |

8 | 1 | 16983563041 | 25600000000 | 4.58E+13 | 1.11E+14 | 1.68E+15 | 2.04E+15 | 3.88E+15 | 1.28E+13 |

n | 2 | 12 | 31 | 40 | 69 | 71 | 85 | sum | |

1 | 2 | 12 | 31 | 40 | 69 | 71 | 85 | 310 | |

2 | 4 | 144 | 961 | 1600 | 4761 | 5041 | 7225 | 19736 | |

3 | 8 | 1728 | 29791 | 64000 | 328509 | 357911 | 614125 | 1396072 | |

4 | 16 | 20736 | 923521 | 2560000 | 22667121 | 25411681 | 52200625 | 103783700 | |

5 | 32 | 248832 | 28629151 | 102400000 | 1564031349 | 1804229351 | 4437053125 | 7936591840 | |

6 | 64 | 2985984 | 887503681 | 4096000000 | 1.08E+11 | 1.28E+11 | 3.77E+11 | 6.18E+11 | |

7 | 128 | 35831808 | 27512614111 | 1.64E+11 | 7.45E+12 | 9.10E+12 | 3.21E+13 | 4.88E+13 | |

8 | 256 | 429981696 | 8.53E+11 | 6.55E+12 | 5.14E+14 | 6.46E+14 | 2.72E+15 | 3.89E+15 | |