36,810 reputation
352125
bio website davidlowryduda.com
location Providence, RI
age 25
visits member for 3 years, 2 months
seen 3 hours ago

I'm working on my Math PhD at Brown University. I've finished my second year, and now I pursue my interests in analytic number theory. In particular, I study automorphic forms under Dr. Jeff Hoffstein.

I happen to loosely update a math blog at davidlowryduda.com. I put a lot of MSE things on there too, though a lot of the material caters to whatever class I'm teaching at the time (this fall, calc I).


1d
comment Finding formulas for sums
@AdamHughes: Ah, I was actually thinking you were referring to the $\phi(d)$ sum, for whatever reason. Sorry about that
1d
comment Prove that a function is solution of IVP
Your last line is confusing. When you plug in $0$ for $x$, you should have no $x$s left.
1d
comment Use the Chinese Remainder Theorem to show that an integer $a$, with $0 \leq a < m = m_1*m_2* \dots *m_n$, …
You say to use the CRT, but I would say this is exactly the CRT on the integers. But your proof is correct.
2d
comment Counting maximum moves
This is part of an ongoing contest, and has been locked until the contest ends.
2d
accepted Elementary, direct proof of when $5$ is a quadratic residue mod $p$
2d
comment Elementary, direct proof of when $5$ is a quadratic residue mod $p$
By the way, I like this proof very much. +1 and (very likely) an accept later. This makes me wonder about something else: now the fact that we can get $-1 \pmod 5$ seems special to the size of the group of units of $F_{p^2}$. Do you see a way to extend this to handle something like... $p \equiv 2 \pmod 7$ (and $\equiv 1 \pmod 4$) leading to $7$ being a quadratic residue mod $p$? Or have we reached the end of how far we can push?
2d
comment Elementary, direct proof of when $5$ is a quadratic residue mod $p$
Silly question from me: why does $\text{GF}(p^2)$ have an element of order $5$? The order of the group of units of $\text{GF}(p^2)$ is $p^2 - 1 = (p+1)(p-1)$... aha. To be honest, I was thinking it was $\varphi(p^2)$ until just now. I'm going to leave this comment here as a note to myself. Next question: $\omega + \omega^p$ is in $F_p$. One way to see this is that it's fixed under Frobenius. Is there a non-Galois theoretic way of seeing it too (that's likely far simpler, but I don't deal very much with these so much anymore)?
2d
comment Elementary, direct proof of when $5$ is a quadratic residue mod $p$
@Geoff: I see this coming from the group of units for $\mathbb{Z}/p\mathbb{Z}$ being cyclic of order $3k$, so that there is an element of order $3$. Then this element satisfies $x^2 + x + 1 \equiv 0 \pmod{p-1}$, and so on. Do you happen to know if Gauss's ideas extend to $p \not \equiv 1 \pmod p$?
2d
comment Elementary, direct proof of when $5$ is a quadratic residue mod $p$
Fortunately, I have my copy sitting right next to me. I'll pull it out and see if this leads me somewhere. Thanks -
2d
comment Elementary, direct proof of when $5$ is a quadratic residue mod $p$
I've looked at some of his writings, but I admit that I have neither seen all of them nor understood everything I read. Most presentations of Gauss's material rely on either Gauss's Lemma or Gauss sums - both of which I know, but want to avoid for now.
2d
asked Elementary, direct proof of when $5$ is a quadratic residue mod $p$
Jul
7
comment Evaluating $\dfrac{1}{\Gamma (r)}\int_{0}^{x}(x-t)^{\alpha -1}t^{\lambda}dt$
You're a short substitution away from the beta function
Jul
7
comment Is there a quick way to obtain $a,b$ in $ax+by = z$ where $x,y,z$ are fixed and $x+1 = y$?
Once you have one solution, you have all solutions. Adding x to a and subtracting y from b will give another.
Jul
7
answered Is there a quick way to obtain $a,b$ in $ax+by = z$ where $x,y,z$ are fixed and $x+1 = y$?
Jul
3
comment Peculiar numbers
In fact, you can also write a recurrence relation that gives you the "magical" numbers of a longer length. I wrote an answer about this type of number before. They're called "automorphic numbers."
Jul
3
answered Dirichlet convolution for dummies
Jul
2
answered Integral test for convergence: $\sum _1^\infty \frac{e^{1/n}}{n^2}$
Jul
2
comment Integral test for convergence: $\sum _1^\infty \frac{e^{1/n}}{n^2}$
What are you integrating, and with respect to what? What is $n$? What are the bounds of integration?
Jul
2
comment Is 39 moves the longest a chess game can go moving only pawns?
There is a puzzling stackexchange too. I'm uncertain whether this would be a better fit there or not.
Jul
2
comment function meromorphic on C
Extended discussion should take place in the chat room instead of in the comment sections to questions. If you want to continue your discussion, I recommend you take it to one of the chat rooms