Problem 318
Posted: Sun Jan 23, 2011 11:36 pm
For Problem 318 (View Problem), I have a bug somewhere. If someone can answer the following two questions, I hopefully can track it down.
Is N(2, 3) 2020? (It actually would be more helpful to do this for a value of (p, q) that converges more slowly. I think I have an algebra error in this calculation.)
For $1 \leq p < q \leq 100$, are there 1185 pairs such that the fractional part of $(\sqrt{p} + \sqrt{q})^{2n}$ converges to 1 as $n$ goes to infinity?
<b>Update:</b> Got it. I fixed all of my algebra errors, but was misreading the condition as $1 \leq p < q \leq 2011$ when it was really $1 \leq p < q$ and $p + q \leq 2011$. If you modify the second question to have the same form as the actual condition, then there are 441 pairs. (Hopefully this is useful to someone else. I'm leaving this up because I don't think this is a spoiler.)
Is N(2, 3) 2020? (It actually would be more helpful to do this for a value of (p, q) that converges more slowly. I think I have an algebra error in this calculation.)
For $1 \leq p < q \leq 100$, are there 1185 pairs such that the fractional part of $(\sqrt{p} + \sqrt{q})^{2n}$ converges to 1 as $n$ goes to infinity?
<b>Update:</b> Got it. I fixed all of my algebra errors, but was misreading the condition as $1 \leq p < q \leq 2011$ when it was really $1 \leq p < q$ and $p + q \leq 2011$. If you modify the second question to have the same form as the actual condition, then there are 441 pairs. (Hopefully this is useful to someone else. I'm leaving this up because I don't think this is a spoiler.)