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Removed the following claim, on the grounds of utter falsehood:
53 (base 10) is a prime number, so are 110101 (binary), and 41 (base 13). Likewise 52 is equal to 2*2*13, while 110100 = 10*10*1101 (binary) and 40 = 2*2*10 (base 13) and so they are not prime numbers. Multiplication and division give the same results in any base, we just write them differently.
Anyway, communication with an unknown race wouldn't be using human numeral systems since the aliens wouldn't know how we write. The simplest way to write out the prime numbers would be something like:
Except that the whole thing relies on aliens 'recognising' dividing prime numbers as getting results that are 'untidy'. Many a mathematics teacher's been annoyed by those students who, when explained that prime numbers are special because they can't be divided by anything except 1 and themselves, maintain that "sure they can, you just get something behind the decimal point".
If the aliens in question use a type of notation or follow a school of thought where decimals are accounted for without such a clearly defined separation marker as a decimal point, their prime numbers might be very different. (for instance, they might consider 1/10, 1/9, 1/8, 1/7, 1/6, 1/5, 1/4, 1/3, 1/2 and 1 to be the first 10 natural numbers, in which case 3 isn't a prime number at all (and in fact, in their notation, there aren't nearly as many prime numbers, if any).
This seems strange to us, because we like to think in whole numbers, how would you even represent that in those pulses you wrote out? Why such a system might even be said to be completely alien.
Note: I did not add the original claim.
Such a system would be not only completely alien, but also rather incoherent. Why is 1/10 a natural number but not 1/11? If 1/10 and 1/9 are both natural numbers, why isn't 19/90 (their sum) or 1/90 (their product) also a natural number? It's certainly possible that aliens would have their own incompatible sense of mathematical interestingness that doesn't care about things like that, but you can say that about any attempt at communicating with them. The argument in favor of First Contact Math isn't that it's foolproof; it's that it relies on fewer assumptions than just about anything else, so it's more likely to succeed.
To expand on Micah:
1,2,3,5,7,11, etc items look the same no matter what base you count in. 7 rocks (or pulses or what ever) are always indivisible into smaller equal groups. If the ET says send pulses in groups of 1,2,3,11,13,23, we will receive X XX XXX XXXXX XXXXXXX XXXXXXXXXXX
and some one will recognise these as groups of Primes
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How well does it match the trope?