# Recent content by math_grl

1. ### Median, average question

nevermind i found it. 25/4 was the last one.
2. ### Median, average question

Let f(x) be the mean of five numbers: 4, 5, 7, 9 and x. Let g(x) be the median of the same numbers. For how many values of x, a real number, is f(x) = g(x)? I only got 2. x = 0, 10 There are 3 though. Perhaps someone can help me find the other one.
3. ### Infinite number of units

Thanks to Petek I was able to clear up my misunderstanding.
4. ### Infinite number of units

Yes Petek you hit the nail on the head, actually the book I'm reading seems to have no details and the few resources I looked up online just said we can derive some particular unit that if you raise it to the nth power, it's still a unit, from the pell's equation. I know the fundamental unit for...
5. ### Infinite number of units

How do you find an infinite number of units of \mathbb{Q}(\sqrt{21}) using the \sqrt(21)? I saw one example using continued fractions but do not know how to apply it in this case. I do have the periodic form of the continued fraction of \sqrt(21).
6. ### Unique factorization domain

I guess it could be. So you are saying that you take some random element a + b\sqrt{5} \in \mathbb{Q}(\sqrt{5}) and claim there are two distinct prime factorizations and show they actually differ by a unit?
7. ### Unique factorization domain

So you see it all over the place, \mathbb{Q}(\sqrt{-5}) is not a UFD by finding an element such that it has two distinct prime factorizations...but what about showing that \mathbb{Q}(\sqrt{5}) is a UFD? I'm only concerned with this particular example, I might have questions later on regarding a...
8. ### Gauss lemma extension?

Ok, the book I'm reading states Gauss's lemma as such: If f(x) is a monic polynomial with integral coefficients that factors into two monic polynomials with coefficients that are rational, f(x) = g(x)h(x), then g(x), h(x) \in \mathbb{Z}[x]. Now one of the exercises says to prove that: If...
9. ### Continued fractions

:blushing: that's embarassing.
10. ### Continued fractions

Ok I need to know which is the right answer for evaluating the continued fraction \langle 1, 2, 1, 2, \ldots \rangle? Here's my work: x = 1 + \frac{1}{2+x} \Rightarrow x^2 + x - 3 = 0 and by quadratic formula, we get x = \frac{-1 \pm \sqrt{13}}{2} but we only want the positive root so I...
11. ### Mobius inversion?

Now you've given me something to think about... very interesting way of proving it as I thought being in the same section as that of Mobius inversion, was supposed to follow direct from that or something...
12. ### Mobius inversion?

\sigma _0 (n) = \sum_{d \mid n} 1 and \omega (n) = \sum_{p \mid n} 1 where p is a prime and d is a divisor. I thought the notation was standard for arithmetic functions.
13. ### Mobius inversion?

Would like to show \sum_{d \mid n} \mu (d) \sigma_0 (d) = (-1)^{\omega (d)}. This proof is just left out of text I'm looking at and I can't seem to piece how F(n/d) = \sigma_0 (d), where F(x) = \sum_{s \mid x} f(x).
14. ### Completely multiplicative

So you are saying that F(4) - F(2)F(2) is not 0 as it should be? I was hoping for more an explicit function, f, such that F is not completely multiplicative.
15. ### Completely multiplicative

If f is completely multiplicative, then \sum_{d \mid n} f(d) is completely multiplicative is not true. There must be an easy counterexample for this yet I cannot come up with one.