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visits | member for | 3 years, 1 month |
seen | Aug 25 at 2:25 | |
stats | profile views | 607 |
God geometrizes continually. But I prefer doing number theory.
Aug 24 |
answered | A family Mersenne composite numbers? |
Aug 12 |
comment |
Divisor sums over values of binary forms of primes
@Gerry: Computing asymptotics for the partial sums of $\tau(p+a)$, with $a$ fixed, is known as the Titchmarsh divisor problem. This was solved by Linnik, but nowadays can be done by the Brun--Titchmarsh and Bombieri--Vinogradov theorems. As for $\tau(x^2+y^2)$, it's maybe more natural to consider the sum extended over all pairs $x,y$ with $x^2+y^2 \le n$. That can be attacked by applying a mean-vale theorem of Wirsing to the function $n\mapsto \tau(n) r(n)$, where $r(n) = \frac{1}{4} \#\{(x,y):x^2+y^2=n\}$. (Note that $r$ is multiplicative.) |
Aug 10 |
comment |
How many integers divide a number that involves just three non-zero digits?
Not an answer to your question, but Skalba has shown that if the order of 2 mod p exceeds $p^{4/5}$, then p divides some number of the form $2^a + 2^b+1$. We don't know that Skalba's criterion holds for a full density of primes of $p$, but we expect it to do so (e.g., that follows from GRH). He also shows that if $2^m-1$ has fewer than $\log{m}/\log{3}$ prime divisors (counted with multiplicity), then there is a prime dividing $2^m-1$ not dividing any number of the form $2^a+2^b+1$. See: Skałba, Mariusz, Two conjectures on primes dividing $2^a+2^b+1$, Elem. Math. 59 (2004), no. 4, 171–173. |
Jul 24 |
comment |
Examples of famous 'workhorse' theorems
Surely "Brun's sieve" qualifies. |
Jul 22 |
comment |
Constructing quintic number fields with certain splitting behaviour
Doesn't the hypothesis in Theorem 1.3 only require what you said for all large enough p? So why can't you (or can you?) run your argument up to an arbitrarily large finite height to get an arbitrarily small upper density, and hence a density of 0? |
Jul 21 |
revised |
Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions
corrected spelling of "specimen" |
Jul 21 |
answered | Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions |
Jul 18 |
comment |
Proof of equidistribution theorem for exponential coefficients
Asking that $\{2^n \alpha\}_{n \ge 1}$ be equidistributed mod $1$ is exactly equivalent to asking that $\alpha$ be normal in base $2$. Not a single irrational algebraic number is known to be $2$-normal; for some partial results, see this paper of Bailey, Borwein, Crandall, and Pomerance: crd.lbl.gov/~dhbailey/dhbpapers/algebraic.pdf |
Jul 17 |
awarded | Enlightened |
Jul 17 |
awarded | Nice Answer |
Jul 17 |
revised |
How frequently is 3 a cubic residue mod primes in an arithmetic progression?
edited body |
Jul 17 |
comment |
How frequently is 3 a cubic residue mod primes in an arithmetic progression?
Thanks Filippo. That's what I get for not reading the question carefully! Edited. |
Jul 17 |
revised |
How frequently is 3 a cubic residue mod primes in an arithmetic progression?
corrected "2" to "3"; fixed some typos |
Jul 17 |
answered | How frequently is 3 a cubic residue mod primes in an arithmetic progression? |
Jul 17 |
awarded | Yearling |
Jul 1 |
comment |
Covering a set with geometric progressions
@Sean Eberhard: See also this very recent preprint of Nathan McNew: nathanmcnew.com/GPFsets.pdf |
May 31 |
comment |
Computing $\prod_p(\frac{p^2-1}{p^2+1})$ without the zeta function?
I agree with Greg's encouragement and suggest also sending a copy to Richard Guy at the same time (if you haven't already emailed him). |
May 30 |
awarded | Good Answer |
May 23 |
comment |
Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$
This doesn't answer your question, but ... you might be interested in knowing that for rather specific families of polynomials $P$ (rather than all $P$ with bounded height), Bardestani has some density results: arxiv.org/abs/1202.2047 |
May 14 |
revised |
Commuting limits in relating the harmonic series to coprimality densities
typos corrected |