Like powers

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