Each bacterium grows at a some constant rate, i.e. every minute the size of the bacteria increases by some constant value. Different bacteria can grow at different rate (they can also grow at same rate). Scientists observe $n$ bacteria under the microscope. Bacteria differ only in size and growth rate (in all other they are indistinguishable). Every minute, scientists identify the sizes of all $n$ bacteria and write down $n$ numbers - the sizes of bacteria (Scientists do not know which particular bacterium is of this or that size, because the bacteria move all the time). Prove that there is a number $m$ such that after $m$ minutes of observations, scientists will be able to unequivocally find a set of the growth rates of $n$ bacteria (The number $m$ should not depend on the sizes of the bacteria and their growth rates. The number $m$ may depend on the number $n$.).
My work. Scientists have numbers of $n$ arithmetic progressions. They need to find a set of common difference of arithmetic progressions. I proved that: a) if $n=1$ then $m=2$; b) if $n=2$ then $m=3$; c) if $n \ge 3$ then $m \ne 3$ (I do not know how to prove of the problem in the case when $n \ge 3$).