<s>[INST] Consecutive sets of consecutive numbers which add to the same total I'm looking at examples of numbers that can be written as the sum of integers from $j$ to $k$ and from $k+1$ to $l$.  For example $15$ which can be written as $4+5+6$ or $7+8$.  Or $27 = 2+3+4+5+6+7 = 8+9+10$.  I have been able to find a few numbers which have two ways to satisfy the above equations.  For example,
$$\begin{aligned}105 
&= 1+2+\dots +14 = 15+16+\dots+20\\
&= 12+13+\dots+18 = 19+20+\dots+23
\end{aligned}$$
However, I have not been able to find any numbers that can be written as the sum in three ways of consecutive sums.  That is, I have not been able to find an $X$ such that,
$$\begin{aligned}X 
&= (a+1)+(a+2)+\dots +b = (b+1)+(b+2) +\dots +c\\
&= (d+1)+(d+2)+\dots +e = (e+1)+(e+2) +\dots +f\\
&= (g+1)+(g+2)+\dots +h = (h+1)+(h+2) +\dots +i\\
\end{aligned}$$
Does any such number $X$ exist?  If so can you provide an example?  If no such number exists can you provide a proof?
Thanks
 [/INST] Suppose, 
$$X=(a+1)+(a+2)+\dots+b=(b+1)+(b+2)+\dots+c$$
which implies,
$$\Leftrightarrow \frac{b(b+1)}{2}-\frac{a(a+1)}{2}=\frac{c(c+1)}{2}-\frac{b(b+1)}{2}$$
$$2b(b+1)=a(a+1)+c(c+1)$$
or the special Pythagorean triple,
$$(2a-2c)^2+(2a+2c+2)^2=(4b+2)^2$$ 
where $a,b,c\in\mathbb Z$. We need to find integer solutions of the system,
$$(2c-2a)=s^2-t^2,\quad (2a+2c+2)=2st,\quad (4b+2)=s^2+t^2$$
So,
$$a=\frac{|s^2-t^2-2st|-2}{4},\quad b=\frac{s^2+t^2-2}{4},\quad c=\frac{2st+(s^2-t^2)-2}{4}$$ 
and just let $s=(2m+1),t=(2n-1)$, where $m\geq n$. Since,
$$X=\bigg(c+\frac{1}{2}\bigg)^2-\bigg(b+\frac{1}{2}\bigg)^2=\bigg(\frac{2st+(s^2-t^2)}{4}\bigg)^2-\bigg(\frac{s^2+t^2}{4}\bigg)^2\\
=\frac{st(t+s)(s-t)}{4}=(2m+1)(2n-1)(m-n+1)mn$$
If the equation below has 3 or more integer solutions then the $z$ is exactly what you want:
$$z=(2m+1)(2n-1)(m-n+1)mn$$ 
where $m\geq n$. Alternatively,
$$\begin{aligned}
a &= \tfrac{1}{2}\Big(-1+\sqrt{(x^2-2xy-y^2)^2}\Big)\\ 
b &= \tfrac{1}{2}\big(-1+x^2+y^2\big)\\
c &= \tfrac{1}{2}\big(-1+x^2+2xy-y^2\big)
\end{aligned}$$
where $x>y$ and sign chosen so $a$ is positive. Then,
$$X = \frac{b(b+1)}{2}-\frac{a(a+1)}{2}=\frac{c(c+1)}{2}-\frac{b(b+1)}{2} =\tfrac{1}{2}xy(x^2-y^2)$$
so it suffices to find three pairs of $x,y$ with the same $X$.  
</s>