mutually assured destruction (MAD) is the idea that 2 countries at nuclear war–like the USA and Russia–could wipe each other out with nuclear weapons. but the situatiuon is much simpler than this.
on the average 1 rocket out of 24 fails. this means it blows up, falls early, and so forth. it turns out that 1 in 24 shuttle launches have been failures as well.
so imagine thousands of rockets sent up to destroy ones foe. for each 1000 rockets 42 of them fail, falling to earth and spreading nuclear waste and poison on the country that launched it.
conclusion: nuclear war doesn’t even need another country’s bombs to destroy the nuclear country. it will destroy itself.
Fermi and Von Neumann suggested the galaxy should be completely populated by aliens if alien advanced civilizations exist, because they could build machines (Von Neumann machines) that would travel in interstellar space, find usable materials, duplicate themselves, and continue to expand in number and in space until the galaxy as full of them. since the process is exponential Von Neumann suggested it should take less than 100 million years, which is a short time compared to the age of the galaxy. Since we don’t see such machines, they must not exist, so advanced alien civilizations must not exist.
But there is a problem. Chemical rockets can only escape planets a little bit bigger than earth, because the greatest chemical rocket specific impulse (lithium and fluorine!) is not high enough to even lift the rocket fuel away from the pull of gravity. look at the payload of the Saturn rocket compared to the rocket size–it is 98% fuel. if a planet were 20-30% more massive than earth no chemical rocket could escape, so the alien advanced civilization would be planet-bound.
of course there is also a minimum size planet that can support sentient life, and sentient life that can build rockets need an insulating atmosphere (see The Anthropic Cosmological Principle, Barrow et al.). This minimum size is not much smaller than earth. so civilizations that can do interplanetary and galactic travel are a very small fraction of such civilizations.
this proof is typically pretty messy, but here is a simple one.
for explanatory purposes let k=2. for any p there are 2^p numbers p digits in length. we will call a rotation set the set of p digit numbers that are formed by taking the last digit of a number and transferring it to the first digit, for example if p=5 one rotation set is 10111, 11011,11101,11110,01111. if p is not prime, then some rotation sets are less than p elements long, because there are repeats of patterns. for example if p=4 then 1010,0101,1010,0101 repeats so this rotation set has less than p elements.
however if p is prime then no such short rotation sets exist, because if by rotating some number of places you could get an element already in the set then the number of rotation steps you did would have to divide p, but p is prime. so if p is prime all the rotation sets have the same number of elements, p.
except this is not true for k sets, 1111…11 and 000..00 each have only one member in the example case.
there are (2^p-2)/p such rotation sets excluding the 111..11 and 000..00 cases. so this must be an integer.
if k is not 2, then there are k such rotation sets with single members 000..00,111..11,222..22 etc. so there are k^p-k full length rotation sets if p is a prime. therefore (k^p-k)/p must be an integer.
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