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Feb
8 |
answered | If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal |
Jan
30 |
comment |
How many solutions does $\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}=1$ have?
@Ian Agol a) The upper bound of the mentioned paper with the $x_i$ not necessarily distinct is also the best available upper bound for your problem, with all $x_i$ distinct. b) Moreover, if $x_i=x_j$ occurs, one has reduced the number of free variables, so that this case should occur less frequently. Therefore, heuristically, I expect that the two cases $x\leq x_j$ or $x_i<x_j$ have (more or less) about the same number of solutions. (But I do not expect that the currently known upper bounds are very close to the true growth function) |
Jan
29 |
comment |
How many solutions does $\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}=1$ have?
The paper by Konyagin (mentioned above by Ian) gives the best known lower bound. The best known upper bound is in a paper by Browning and Elsholtz, The number of representations of rationals as a sum of unit fractions, Illinois J. Math. 55, Number 2 (2011), 685-696. maths.bris.ac.uk/~matdb/preprints/IJM343.pdf |
Jan
20 |
revised |
Small quotients of smooth numbers
added 1085 characters in body |
Jan
14 |
revised |
Small quotients of smooth numbers
commented on $C(k)$ |
Jan
14 |
revised |
Small quotients of smooth numbers
added 118 characters in body |
Jan
14 |
answered | Small quotients of smooth numbers |
Dec
14 |
comment |
Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$
In the integer case, with $n=4$ there is some info here, on a disproved conjecture of Euler, en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture Elkies had found examples. For $n\geq 5$ examples are known if one uses more variables. (I guess with 3 summands, $n\geq 5$ nothing is known) |
Sep
20 |
revised |
Upper bound on length of addition chain
added 386 characters in body |
Sep
20 |
revised |
Upper bound on length of addition chain
added 1565 characters in body |
Sep
18 |
comment |
Upper bound on length of addition chain
Yes, some minor correction just added. $C$ is close to 1, for quite large values of $n$. |
Sep
18 |
revised |
Upper bound on length of addition chain
Corrected. |
Sep
18 |
answered | Upper bound on length of addition chain |
Sep
1 |
revised |
Lower bounding the multiplicative order of 2 modulo p
added 1295 characters in body |
Aug
31 |
revised |
Lower bounding the multiplicative order of 2 modulo p
(edited jstor link). |
Aug
31 |
answered | Lower bounding the multiplicative order of 2 modulo p |
Jul
7 |
awarded | Yearling |
May
22 |
revised |
For $k>3$ does there exist an odd prime $q_k$ such that $p_k=2^kq_k+1$ is prime and $p_k$ divides $a_k=\dfrac{3^{2^{k-1}}+1}{2}$?
added 762 characters in body |
May
22 |
answered | For $k>3$ does there exist an odd prime $q_k$ such that $p_k=2^kq_k+1$ is prime and $p_k$ divides $a_k=\dfrac{3^{2^{k-1}}+1}{2}$? |
Aug
24 |
awarded | Nice Answer |