Impact
~136k
people reached
- 0 posts edited
- 0 helpful flags
- 96 votes cast
Dec
27 |
comment |
Every prime number > 19 divides one plus the product of two smaller primes?
Mark - yes, there are "sum-product" results of this type. The one most applicable here is that due to Bourgain-Katz-Tao: it says that if $A \subseteq Z/pZ$ and $|A + A|, |A \cdot A| \leq K|A|$ then either $|A| \leq K^C$ or $|A| \geq K^{-C} p$. When $A$ has size about $p/\log p$, it tells you rather little (namely either $|A + A|$ or $|A \cdot A|$ has size at least $p/(\log p)^{1 - c}$) and I think exponential sum techniques are likely to be better. |
Dec
27 |
comment |
Every prime number > 19 divides one plus the product of two smaller primes?
I might add that I would expect trigonometric sums to allow one to show that $A \cdot A$ is almost all of $Z/pZ$ as $p \rightarrow \infty$. |
Dec
27 |
comment |
Every prime number > 19 divides one plus the product of two smaller primes?
Interesting problem. I don't think either the sum-product idea or the trig sums idea will work immediately. In the latter case because one has a binary problem instead of one with three variables, and in the former because I think you can have both $A.A \subsetneq Z/pZ^*$ and $A + A^{-1} \subsetneq Z/pZ$ with $A$ reasonably large, even positive density. But both of these are good ideas on which to base further thought. |
Dec
19 |
awarded | Notable Question |
Dec
16 |
comment |
Picking out an odd subset
I should also add that the second question is rather natural: if I tell you $M(p,h)$ for all $h$ mod $p$ and for all $p \leq X$, how big does $X$ need to be before you can tell me $|M|$? Perhaps I'll ask some of the Bristol number theorists today. |
Dec
16 |
comment |
Picking out an odd subset
Perhaps ideas connected with large sieve will help here. If $M(p,h)$ denotes the number of elts of $M$ congruent to $h$ mod $p$ then the assumption that $M(p,h)$ is always even should force the square mean $\sum_h |M(p,h) - |M|/p|^2$ to be, say, twice the size that one might expect for a positive fraction of $p$s. I'm not claiming to have an actual argument, though! |
Dec
12 |
answered | Is there a Plancherel Theorem for Gowers norms? |
Dec
2 |
comment |
A conjecture on the relative size of Goldbach pairs?
There is most likely a literature on results of the form "For almost all even numbers 2N we can write 2N = p_1 + p_2 with p_1/p_2 < 1/F(N)". I'm pretty sure simply getting F(N) -> infty would be a simple application of the circle method, but I'd be surprised if one could get much better than F(N) > N^c for some small c, if that. |
Nov
26 |
comment |
Minimum number of twin primes < N
Greg, I was tempted to quibble this comment. But then it occurred to me that there really aren't any modern approaches to the twin prime conjecture. One could imagine an approach using nonstandard analysis, though, which would give no quantitative lower bounds. |
Nov
11 |
comment |
A question about groups of intermediate growth
When making my comment above I was addressing the question for "supremum" instead of "infimum". I feel one does need Gromov/Pansu to do that. |
Nov
10 |
comment |
A question about groups of intermediate growth
Nice question. My impression is that there is no simpler proof of 2. You can get some way beyond polynomial growth by quoting Shalom and Tao. |
Oct
25 |
comment |
Arithmetic Progressions of Squares
Kevin - I'd suggest asking one of them. I think there is a preprint. |
Oct
25 |
awarded | Enlightened |
Oct
24 |
awarded | Nice Answer |
Oct
24 |
revised |
Arithmetic Progressions of Squares
added 270 characters in body; added 310 characters in body |
Oct
24 |
answered | Arithmetic Progressions of Squares |
Oct
21 |
awarded | Enlightened |
Oct
20 |
awarded | Nice Answer |
Oct
20 |
comment |
Infimums of exponential sums involving primes
For a long time it was a conjecture of Littlewood that the infimum is $o(\sqrt{x})$. Actually it was Konyagin who got down to $x^{-1/2 + o(1)}$. |
Oct
20 |
revised |
Infimums of exponential sums involving primes
added 385 characters in body; deleted 2 characters in body |