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The statement that "$2$ is the only even prime number" has always struck me as very peculiar. I do not find this statement mathematically interesting, though I do find the fact that it is presented as something interesting about $2$ or prime numbers to be itself quite interesting.

I find it interesting because this statement has secured its place as a "math-tidbit", if you will, solely because we happen to have a word for being divisible by $2$.

In other words, it is not at all clear to me why the statement "$2$ is the only even prime" is any more interesting or worth saying than the statement "$7$ is the only prime that is divisible by $7$."

Every prime $p$ is the only prime divisible by $p$. This immediately follows from the definition. So my question is, what makes the $2$ case particularly special or interesting, other than the fact that we have a word for divisibility by $2$?

Peter Phipps
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Jonathan Hebert
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    It isn't interesting. It is just a consequence of the definition of prime. It is obvious that "most" evens can't be prime, but $2$ is ¯_(ツ)_/¯ – Randall Dec 11 '22 at 23:34
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    Fields of characteristic 2 has some special properties. https://math.stackexchange.com/questions/1573308/whats-so-special-about-characteristic-2 – CyclotomicField Dec 11 '22 at 23:36
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    It's for the same reason that $3$ is the only ternary number that is prime, and the same reason that $5$ is the only quinary number that is prime. And so on. – Accelerator Dec 11 '22 at 23:41
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    There are, however, certain arguments that break down at the prime 2 (see: homotopy theory), but that's a different level of interesting I think. – Randall Dec 11 '22 at 23:43
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    There are at least two very different questions you might be asking: 1. Why does pop math mention that 2 is the only even prime and does that statement have any value? 2. Why is it significant to advanced mathematics that 2 is an even prime/in what ways is 2 notable among primes? – Mark S. Dec 12 '22 at 04:20
  • "interesting" is subjective anyway. But this statement surely isn't interesting , however often used. That $2$ is special in many contexts (reciprocity law , for example) is a completely different story. – Peter Dec 12 '22 at 09:41
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    Maybe this is an old one, but I still like it: "all primes are odd, except $2$ - this is the oddest of all." – Gottfried Helms Dec 12 '22 at 10:47
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    Look at it the other way: why do we have words for odd and even? I think the answer is more psychological than mathematical. We characterize our experiences dichotomously: good/bad; light/dark; up/down; etc. The importance of two-ness precedes any mathematical investigations, so when we start to think mathematically, we transfer our notion of the importance of $2$ and come up with concise words for those quantities (numbers) that are two-like and those that are not. Then we observe that our division of numbers into those categories leaves only one prime number in the two-like category. – Keith Backman Dec 12 '22 at 14:50
  • @KeithBackman very insightful, thank you. I often wonder how much the human experience shapes our mathematics. People like to say that mathematics is "universal", and in a sense that is true, but as mathematics is one of the most creatively free and unrestrained topics there are, I suspect that our humanity influences many things, such as what we decide to designate as axioms vs what we designate as theorems, what objects we choose to define, etc. I suspect some alien civilization's mathematics would be unrecognizable to us, just through the creative choices that math affords us. – Jonathan Hebert Dec 16 '22 at 06:41

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You are right that the quoted statement as such loses its charm upon a moment of consideration. After all, as you say, "$2$ is the only even prime" is not more or less surprising than "$7$ is the only prime divisible by $7$".

One could think of it this way: Why is $2$ the only prime $p$ such that we have a special word for "divisible by $p$"? -- That is a question more about language than about math, but it hints at the fact that the number $2$ holds a more special place in our psychology than all other natural numbers (although some of the other primes, especially $3, 5, 7,$ and $13$, stand out too, in some dominant cultures. $57$, not so much.)

Now, this site is not about psychology, culture, or language. But even mathematicians will sometimes utter a statement similar to the one you quote. What could they mean?

It turns out that in various theories, the prime $p=2$ (and to a lesser but notable extent, $p=3$) behaves differently than all the other primes! See: https://mathoverflow.net/q/160811/27465, and compare https://mathoverflow.net/q/915/27465. Also, the whimsically phrased Why are even primes notable?. A basic example that sets aside $p=2$ (which causes many others) is that there are two rational solutions to $x^2=1$, while for all other $p$, the equation $x^p=1$ has only one real solution. (For more advanced people: The unit group of $\mathbb Z$ is $ \simeq\mathbb Z/2$. Or: The only primitve roots of unity in $\mathbb R$ are $\pm 1$.)

Now the statement "$2$ is the only even prime" might be an exceptionally bad way to express this phenomenon. More fitting might be the old joke, as per Gottfried Helms' comment: "$2$ is the oddest prime."

Torsten Schoeneberg
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  • "Even" means divisible by 2, and 2>1, so there could not be more than one even prime. – DanielWainfleet Dec 12 '22 at 19:06
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    I assume you've seen this, but, just in case, for your entertainment - from John Tate, at the FLT conference a few hundred years ago: https://youtu.be/lbsniAHLY4Q?t=1457, at least until some 90 secs after, with '2-1=1' , but also a few seconds past https://youtu.be/lbsniAHLY4Q?t=1894. (And what's up with this disparagement by you of the much esteemed Grothendieck prime?) – peter a g Dec 12 '22 at 19:16
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    Also, while Tate and 2 are on my mind: "During a seminar at Harvard, a conjecture of Lichtenbaum's was mentioned. Someone scornfully said that for the only case that anyone had been able to test it, the powers of 2 occurring in the conjectured formula had been computed and they turned out to be wrong; thus the conjecture is false. "Only for 2" responded Tate from the audience. [And, in fact, I think the conjecture turned out to be correct except for the power of 2.]" (from https://www.jmilne.org/math/apocrypha.html) – peter a g Dec 12 '22 at 19:23
  • Downvoter, please explain what's the issue? – Torsten Schoeneberg Dec 15 '22 at 18:31
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    +1 for Grothendieck prime – Reinstate Monica Dec 15 '22 at 19:55
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    Thank you for taking the time to write such a thoughtful answer. I agree that reframing the thought, and shifting away from the "exceptionally bad way to express this phenomenon" allows us to appreciate this little toy topic much more. I enjoyed reading through your thoughts, and the x^p = 1 example was very insightful. – Jonathan Hebert Dec 16 '22 at 06:33