# Solving an integer equation (equi-energy transition)

In chemistry, we came across an equation as follows:

$$\frac{Z_1^2}{n_1^2}-\frac{Z_1^2}{n_2^2}=\frac{Z_2^2}{n_3^2}-\frac{Z_2^2}{n_4^2}$$

We were supposed to assume that this implied that

$$\frac{Z_1}{Z_2}=\frac{n_1}{n_3}=\frac{n_2}{n_4}$$

Is there any mathematical reason why this holds true, or is it just a result that usually (but not always) holds true. All the $6$ variables mentioned are small positive integers (small would mean less than $100$ for sure).

• Infinitely many solutions. And not only these. math.stackexchange.com/questions/153603/… – individ May 28 '16 at 11:36
• Not always. Take $Z_1=5,Z_2=1,n_1=48,n_2=36,n_3=16,n_4=9$. – mathlove May 28 '16 at 11:53
• @individ If the variables are smaller than 100 and greater than 1, surely there are not infinitely many solutions. – YoTengoUnLCD May 28 '16 at 11:57