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Neither is prime (Posted on 2014-12-30) Difficulty: 2 of 5
Show that both 10101 and 10201 are composite in any base b, b>2.

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

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Some Thoughts One out of two ain't bad | Comment 1 of 2

By definition (10101)base b = b^4 + b^2 +1

            and (10201)base b = b^4 + 2b^2 +1

Now b^4 + 2b^2 +1 = (b^2 + 1)^2 so it's composite for all real b.

But b^4 + b^2 +1 is a quadratic in b^2 which can't be factored in real numbers.

However, an Excel test up to b = 562 yields only composite integers for b^4 + b^2 +1. But how to prove it?

  Posted by JayDeeKay on 2014-12-30 10:35:53
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