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Dec
2
comment Is Lehmer's polynomial solvable?
@IgorRivin I agree, it's a bit curious. I could presumably construct a clumsy proof that it's not the full hyperoctahedral, but I wonder if there's a better reason. (I am inclined to trust Magma.)
Dec
1
comment Is Lehmer's polynomial solvable?
Magma quickly computes that it's a nonsolvable group of order 1920.
Nov
24
awarded  Nice Answer
Nov
24
awarded  Yearling
Nov
22
awarded  Critic
Nov
22
answered BSD and congruent numbers
Nov
18
awarded  Nice Answer
Nov
18
answered Primes $P_{2n-1}$ that are $2$ mod $3$
Feb
5
comment Is there a Poisson Summation formula for imprimitive Dirichlet characters?
You might be interested in a preprint of Daileda and Jones, where they show that by modifying the way in which one extends primitive characters to imprivitive characters (in particular, by making a choice other than $\chi(n)=0$ for $n$ not coprime to $q$ -- and, iirc, by choosing it so that the Gauss sum is well-behaved), these new imprimitive characters behave nicely analytically. It's available here: olemiss.edu/working/ncjones/primitivity9.pdf .
Dec
16
answered Least supersingular prime
Nov
24
awarded  Yearling
May
15
answered S integral points of an elliptic curve, with S of positive density
May
4
answered A square-squareroot integer race sequence involving primes
May
3
comment A square-squareroot integer race sequence involving primes
Legendre conjectured that there is always a prime between $n^2$ and $(n+1)^2$, and this is generally believed to be true. Assuming this, for any $n$, $g(f(n))=n$, so you're just looking at a random walk.
May
3
comment A question on primitive roots
There's surely a dependence on the choice of the primitive roots (assuming my interpretation of the question is correct). Look at the primes $q$ for which $x_p \equiv a\pmod{n}$, and then choose a different primitive root for each such $q$. Why should the congruence $x_p\equiv a\pmod{n}$ be preserved for almost all of these $q$? It might also be possible to hack together a sequence of $x_p's$ which are not equidistributed modulo any $n$, but I'm too tired to think through the details of this right now.
May
3
comment A question on primitive roots
Seva: Despite the notation, it seems that $q$ varying is more interesting. If we fix $q$ and vary $p$, Dirichlet's theorem shows that each possible value of $x_p$ occurs equally often.
May
2
comment Is the alternating sum of primes $2 - 3 + 5 + . . P_N$ asymptotic to $- N ln N/2$ for even $N$?
And, I guess, regarding Ruiz's purported proof: I haven't read it, but the discussion suggests that it's replacing $p_n$ by the asymptotic $p_n\sim n\log n$. This cannot work: there are many sequences which satisfy all the properties we know primes to have (e.g., the prime number theorem), and yet are such that, using the notation from my answer, $S(N;a,q)/p_N$ can tend to any value in $[0,1]$, or for which the limit doesn't exist, etc. To prove that $S(N;1,2) \sim p_N/2$, you have to rule out the possibility that all small prime gaps occur with odd index. But I have no idea how to do that!
May
2
comment Is the alternating sum of primes $2 - 3 + 5 + . . P_N$ asymptotic to $- N ln N/2$ for even $N$?
Regarding the generalization with $k\geq 2$: Using essentially the same techniques and that $p_{n+1}^{k+1}-p_n^{k+1}=(p_{n+1}-p_n)(p_{n+1}^k+\dots+p_n^k)=(p_{n+1}-p_n)((k+1‌​)p_n^k + O_k(p_n^{k-1}))$, it would again be possible to prove a lower bound of the right order of magnitude for the alternating sum of $k$-th powers of primes.
May
2
awarded  Nice Answer
May
1
answered Is the alternating sum of primes $2 - 3 + 5 + . . P_N$ asymptotic to $- N ln N/2$ for even $N$?