Question: Here are three sequences . Which numbers are prime? 101, 10101, 1010101, ... 1001, 1001001, 1001001001, ... 10001, 100010001, 1000100010001, ... Hint: Replace each 1

Here arethree sequences.

Which numbers are prime?

101, 10101, 1010101, ...

1001, 1001001, 1001001001, ...

10001, 100010001, 1000100010001, ...

Hint:Replace each 1 with 10 first and then replace each 1 with x and treat each as a polynomial in x. So now the second sequence looks like:

x3+ 1, x6+ x3+ 1, x9+ x6+ x3+ 1, ...

For all of these determine when and how can you factor in Z[x]. If you continue you will run into this question:

When does x4+ x3+ x2+ x + 1 divide x4m+ x3m+ x2m+ xm+ 1?

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