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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
Jul
7
awarded  Yearling