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Like Powers

Age 11 to 14 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