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A spreadsheet was a really useful tool for looking for patterns by quickly calculating powers of each of the numbers and then looking for patterns, but not for explaining why. Why is the power always of the form 2k+1 of 4k+2 except for 5 n + 5 n ?
Here are some combinations (n and k are integers):
1 n + 1 n | never (why?) |
1 n + 2 n | never (why?) |
1 n + 3 n | when n = 4k+2 (why?) |
1 n + 4 n | never |
1 n + 5 n | ever |
1 n + 6 n | never |
1 n + 7 n | when n = 4k+2 |
1 n + 8 n | never |
1 n + 9 n | when n = 2k+1 |
2 n + 2 n | never |
2 n + 3 n | never |
2 n + 4 n | when n = 4k+2 |
2 n + 5 n | never |
2 n + 6 n | when n = 4k+2 |
2 n + 7 n | never |
2 n + 8 n | when n = 2k+1 |
2 n + 9 n | never |
3 n + 3 n | never |
3 n + 4 n | never |
3 n + 5 n | never |
3 n + 6 n | never |
3 n + 7 n | when n = 2k+1 |
3 n + 8 n | never |
3 n + 9 n | when n = 4k+2 |
4 n + 4 n | never |
4 n + 5 n | never |
4 n + 6 n | when n = 2k+1 |
4 n + 7 n | never |
4 n + 8 n | when n = 4k+2 |
4 n + 9 n | never |
5 n + 5 n | when n=k |
5 n + 6 n | never |
5 n + 7 n | never |
5 n + 8 n | never |
5 n + 9 n | never |
6 n + 6 n | never |
6 n + 7 n | never |
6 n + 8 n | when n = 4k+2 |
6 n + 9 n | never |
7 n + 7 n | never |
7 n + 8 n | never |
7 n + 9 n | never |
7 n + 9 n | when n = 4k+2 |
8 n + 8 n | never |
8 n + 9 n | never |
9 n + 9 n | never |