Nitpick, 50 is 2^1 × 5^2, not the other way around. But anyway, thanks a ton for this post, I felt my brain shift at a new way of thinking about numbers, seriously. I have to wonder what serious and useful implementations of this method might have been done somewhere.
By the way, if you're wondering if anyone has done anything useful with the method of describing numbers as prime exponents I mentioned the answer is yes, depending on your definition of "useful".
The encoding in terms of prime exponents is known as the Godel encoding, and its a fairly important step in proving things like Godel's incompleteness theorems and the undecidability of the Halting problem.
Essentially it's a one-to-one map between natural numbers and finite sequences of natural numbers (since you can encode (a,b,c,d...) as 2^a × 3^b × 5^c × 7^d etc), which turns out to be mathematically a convenient thing to have.
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Nitpick, 50 is 2^1 × 5^2, not the other way around. But anyway, thanks a ton for this post, I felt my brain shift at a new way of thinking about numbers, seriously. I have to wonder what serious and useful implementations of this method might have been done somewhere.
By the way, if you're wondering if anyone has done anything useful with the method of describing numbers as prime exponents I mentioned the answer is yes, depending on your definition of "useful".
The encoding in terms of prime exponents is known as the Godel encoding, and its a fairly important step in proving things like Godel's incompleteness theorems and the undecidability of the Halting problem.
Essentially it's a one-to-one map between natural numbers and finite sequences of natural numbers (since you can encode (a,b,c,d...) as 2^a × 3^b × 5^c × 7^d etc), which turns out to be mathematically a convenient thing to have.
Thanks :) I edited my comment so the arithmetic is (hopefully) right now