Hmm... I agree this is compelling. However, since I'm resistant to updating my world view about 1-the-discriminated-prime-number, I'll continue to proffer counter-arguments:
the Fundamental Theorem of Arithmetic is pretty important, but may still not be the "essence" of what prime is
the FTA itself requires the "except 1" clause: "all natural numbers can be uniquely factored into primes except 1" -- which would make someone thing 1 ought to be prime
the FTA already assumes 'modulo permutations', we could easily throw in 'modulo 1'
Wikipedia -- the first and last authority on such things -- carefully writes in an entire sentence unto itself, "The number 1 is by definition not a prime number," suggesting just how arbitrary this is. (My own emphasis added.)
The best argument I came up with for not including 1 as prime, because I tend to worry about how things are constructed, was with the seive of Eratosthenes.
The seive of Eratosthenes says that you can find the primes by starting with all the natural numbers > 1; let 2 be the first prime number, and then begin eliminating all multiples of 2 and the multiples of subsequent primes as you find them. If you included '1' in the first step, then you would eliminate all the numbers in the first step.
Wikipedia -- the first and last authority on such things -- carefully writes in an entire sentence unto itself, "The number 1 is by definition not a prime number," suggesting just how arbitrary this is.
A measure of the arbitrariness is the history, which is that 1 was considered prime up to the 19th century and was a matter of fashion during the 19th century. That suggests that unique factorization is not, in itself, enough to motivate the definition. Perhaps its extension to the gaussian integers or the more radical version for general numbe...
Today I looked at the above illusion and thought, "Why do I keep thinking A and B are different colors? Obviously, something is wrong with how I am thinking about colors." I am being stupid when my I look at this illusion and I interpret the data in such a way to determine distinct colors. My expectations of reality and the information being transmitted and received are not lining up. If they were, the illusion wouldn't be an illusion.
The number 2 is prime; the number 6 is not. What about the number 1? Prime is defined as a natural number with exactly two divisors. 1 is an illusionary prime if you use a poor definition such as, "Prime is a number that is only divisible by itself and 1." Building on these bad assumptions could result in all sorts of weird results much like dividing by 0 can make it look like 2 = 1. What a tricky illusion!
An optical illusion is only bizarre if you are making a bad assumption about how your visual system is supposed to be working. It is a flaw in the Map, not the Territory. I should stop thinking that the visual system is reporting RGB style colors. It isn't. And, now that I know this, I am suddenly curious about what it is reporting. I have dropped a bad belief and am looking for a replacement. In this case, my visual system is distinguishing between something else entirely. Now that I have the right answer, this optical illusion should become as uninteresting as questioning whether 1 is prime. It should stop being weird, bizarre, and incredible. It merely highlights an obvious reality.
Addendum: This post was edited to fix a few problems and errors. If you are at all interested in more details behind the illusion presented here, there are a handful of excellent comments below.