Stable Numbers

Let's begin by considering a few examples. Clearly, the numbers $1$ and $2$ do not satisfy. But the number $3$ satisfies since we can write: $\frac 16 +\frac 12 +\frac 13 =1$

Notice that we can split the term $\dfrac 13$ into two terms $\dfrac 1{12}$ and $\dfrac 14$ . Which imlplies that even $4$ is stable .

Let's carefully observe the splitting that we have performed, $\frac 1{3\cdot 2^0}=\frac 1{3\cdot 2^2}+\frac 1{2^2}$

There we go! We can split write any fraction of the from $\dfrac 1{3\cdot 2^r}$ into two terms as follows: $\frac 1{3\cdot 2^r}=\frac 1{2^{r+2}}+\frac 1{3\cdot 2^{r+2}}$

Hence every number greater than $3$ is stable .

Let k be stable $\sum$ $\frac{1}{a1}$ =1....Now divide by 2...add 0.5...I mean $\sum$ $\frac{1}{2a1}$ + $\frac{1}{2}$ =1...so k+1 is also stable(Note all $2a1$ <2)...Now 1= $\frac{1}{2}$ + $\frac{1}{2}$ (not acceptable since they must be distinct)= $\frac{1}{2}$ + $\frac{1}{3}$ + $\frac{1}{6}$ (acceptable)...So 3 is a stable no. and by induction any no.>3 is stable.

1/6 + 1/2 + 1/3 is clearly a soln.....

So 1/2 + 1/2 *(1/2 + 1/3 + 1/6) must also be a soln.... Similarly again add half to the half of previous expression..... So there are infinitly many soln.... Simple😊