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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
6 months, 3 weeks 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. Sal Gard · 6 months, 3 weeks ago

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@Sal Gard I have another unanswered discussion question. Can you take a look at it? D K · 6 months, 3 weeks ago

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@Sal Gard Nice observation. Kudos. D K · 6 months, 3 weeks ago

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That's interesting. I can show that if \(k\) is composite, then \(a_k\) is composite and has a factor of 101...101. Chung Kevin · 6 months, 3 weeks ago

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@Chung Kevin you mean if k is odd ? then ak will have 101 as a factor. but I also need proof for when k is even. D K · 6 months, 3 weeks ago

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