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May
19
comment Non-standard Gauss sums
Yes indeed! I forgot about my comment!
May
19
comment Non-standard Gauss sums
Isn't Elkies's answer about ruling out the maximal size $2\sqrt{p}$ for Kloosterman sums, rather than zero?
May
16
comment A question on the bounds of the $n$-th composite $c_n$
@user170039: It looks like I was right the first time, and miscalculated the second time. I agree with Gerhard Paseman's answer below, and your question is indeed closely related to the (wrong) Hardy-Littlewood conjecture.
May
15
comment Important open problems that have already been reduced to a finite but infeasible amount of computation
Konyagin not Kolyvagin
May
11
comment Primes p=4k+1 such that k!+1 is divisible by p
@RolandBacher: Note throughout that $a^2+b^2=p$. Note that for example $37=1^2+6^2$, so you can't take $a=-3$ there. (What's being used at the end is that there's essentially only one way to write a prime $p\equiv 1\pmod 4$ as a sum of two squares.)
May
10
comment A question on the bounds of the $n$-th composite $c_n$
I take back my comment and vote. The question is asking something weaker like $\pi(c_{x}) +\pi(c_y) \ge \pi(x+y)$, which would follow from precise enough versions of the prime number theorem.
May
10
comment A question on the bounds of the $n$-th composite $c_n$
I'm voting to close, since this is essentially the well known Hardy-Littlewood conjecture that $\pi(x) +\pi(y) \ge \pi(x+y)$ which is believed to be false.
May
9
comment Are there always at least *five* divisions?
Yes, this also follows easily by just dividing $p$ into the possible residue classes $\pmod 8$, and doing some simple algebra.
May
6
comment Sums of two squares: positive lower density?
I'm voting to close this question as off-topic because this was already mentioned in the linked question.
Apr
30
comment Order of torsion group
@WillSawin: I can count to $1023$ on my fingers -- can't you?
Apr
29
comment Distribution of the sequence $\bigl (\frac{\phi(n)}{n}\bigr )_{n=1}^{\infty}$ in $[0,1]$
The existence of a distribution is well known -- see old works of Schoenberg, and Erdos and Wintner. It is certainly not equidistribution: for example note that all even numbers, and all odd multiples of $105$ will have $\phi(n)/n <1/2$.
Apr
27
comment Is the set $ AA+A $ always at least as large as $ A+A $?
The claim for subsets of the integers holds, and follows from this MO question: mathoverflow.net/questions/168844/…
Apr
26
comment Density of polynomials which are soluble with respect to a set of primes
It's still a tiny bit tricky. What you're thinking of is that a random polynomial, which will have Galois group $S_n$ typically will have this feature as you vary over $p$. But for a fixed prime, there is a positive probability that the discriminant is a multiple of $p$, and this I think will affect some calculations. I didn't think this through fully, so maybe you're fully correct, but it's just not immediate why.
Apr
26
comment Density of polynomials which are soluble with respect to a set of primes
Some qualification of this answer seems needed. The question fixes the degree $n$. For example when $n=1$ or $n=2$ this answer is not correct.
Apr
24
comment Asymptotic limit of truncated Legendre sieve
@user45947: You're right that this is insufficiently well known. For example, I think I should have known this, but I didn't! One lives and learns.
Apr
23
comment Asymptotic limit of truncated Legendre sieve
@TerryTao: Yes indeed it is $1$. I should have thought an epsilon more!
Apr
21
comment Is this theorem on $L$-functions known?
It seems to me that some result like yours would follow from playing around with the Hadamard factorization theorem. I think you should look carefully at the literature. While it may be hard to find exactly what you've stated, you'll find that many related interesting questions, and maybe you'll have ideas on some of them. Good luck!
Apr
20
comment Small quotients of smooth numbers
I think this is a very interesting question, and don't believe it is known. Note that it implies your other question on logarithms of ratios of square-free numbers, which also is unknown I think. Finally, it may be worth pointing out that the kind of bound you are asking for is best possible -- random products of the first $k$ primes will cluster, and then use pigeonhole to find two near each other.
Apr
15
comment When has the Borel-Cantelli heuristic been wrong?
How about the Maier phenomenon that occasionally there are intervals around $x$ of length $(\log x)^{100}$ say with substantially more (or fewer) primes than one would expect?
Apr
14
comment distribution of $\sqrt{-1} \mod p$
Hooley proved the equidistribution of roots $\mod d$ for composite $d$. That is a much easier problem than the one resolved by Duke, Friedlander and Iwaniec.