AoPS Academy
prove (or disprove) that there exists infinitely many positive integers n in which sums to k! where k is a positive integer. (examples, 1+2+3=3! and 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15=5!
has three distinct real roots 
, 
and 
. The polynomial 
has roots ![$\sqrt[3]{\alpha}$](http://latex.artofproblemsolving.com/4/f/5/4f57cc54c79420c509385c423ffa42820cef3109.png)
, ![$\sqrt[3]{\beta}$](http://latex.artofproblemsolving.com/d/4/0/d40f4039c320468d9974a9504917b56eaf73c947.png)
, ![$\sqrt[3]{\gamma}$](http://latex.artofproblemsolving.com/7/8/1/78171ba6cd442d8156fd70b3d845075e3d82a926.png)
. Find the integer closest to 
.
, since we can extend our given equation to 
. Next, note that the range of 
is ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
, so we want to be sure our value of 
gives a value for 
in the range . The only working value is 
, giving 
. We now require 
, and we can solve for 
to get 
.
has the properties that 
for all positive real values of 
for 
.
.
just consists of copies of the path 
"expanded" by 
every time 
is from the closest power of 
)
.
and 
the function reaches a maximum of 
is 
and 
we will have another 
.
.
