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God geometrizes continually. But I prefer doing number theory.


Oct
13
revised Prime factors of the members of a certain recurrence
deleted 12 characters in body
Oct
13
answered Prime factors of the members of a certain recurrence
Sep
24
awarded  Autobiographer
Sep
23
comment Bounds re Asymptotic Formula for the Sum of Largest Prime Factors
Actually [1] predates the Alladi--Erdos paper; see: A. E. Brouwer, Two number theoretic sums, Mathematisch Centrum, Amsterdam, 1974, Mathematisch Centrum, Afdeling Zuivere Wiskunde, ZW 1974.
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