# Prime Sequence

Consider the sequence $$a_1 = 101, a_2 = 10101, a_3 = 1010101$$ and so on. Prove that $$a_k$$ is composite iff $$k\geq 2$$.

Note by D K
2 years, 2 months ago

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Aha I see it. Any even one a sub 2k can be written in the form (a sub k)^2-100^k which neatly factors to (a sub k+10^k)(a sub k-10^k) for all evens. Sorry for my latex. And the odd case is trivial.

- 2 years, 2 months ago

I have another unanswered discussion question. Can you take a look at it?

- 2 years, 2 months ago

Nice observation. Kudos.

- 2 years, 2 months ago

That's interesting. I can show that if $$k$$ is composite, then $$a_k$$ is composite and has a factor of 101...101.

- 2 years, 2 months ago

you mean if k is odd ? then ak will have 101 as a factor. but I also need proof for when k is even.

- 2 years, 2 months ago