Why is 2 the only prime number? What makes 2 special?

From the time we learn about prime numbers, we quickly notice that 2 is the only even prime number. All other prime numbers are odd. So why is 2 so special?

The “specialness” of 2 arises less from the number itself, and more from the limitations of our English language. We have the word even, which means divisible by 2 and we have the word odd which means not divisible by 2. Half of whole numbers are even and the other half odd. Or to put that mathematically, 1/2 of whole numbers are even and 1/2 are odd. Generalizing one step further, which may seem unnecessary at the moment, but the reason for which will become apparent as we continue, (2-1)/2 numbers are odd.  

But we lack words for divisible by 3 and not divisible by 3. So let’s create these words now. Let’s define “theven” as divisible by 3 and “thodd” as not divisible by 3. We can see that 1/3 of numbers are theven and (3-1)/3 or 2/3 of numbers are thodd. Now we can note that 3 is the first and only prime number that is theven. Any number greater than 3 that is theven is not prime because it would be divisible by 3. Or, to put it another way, prime numbers greater than 3 must be thodd. Combined with our prior rule about odd numbers, we can say that prime numbers greater than 3 must be both odd and thodd.

Turning now to 4, 4 is even… so it is not prime.

5 is the next number that is both odd and thodd so it is our next prime. Let’s create the words “feven” and “fodd” for divisible by 5 and not divisible by 5. We can see that 1/5 of numbers are feven and (5-1)/5 or 4/5 of numbers are fodd and only fodd numbers above 5 can be prime. Combining all the rules together, numbers above 5 must be odd, thodd and fodd to be prime.

6 is both even and theven … so definitely not prime.

7 is the next number that is odd, thodd and fodd so it is the next prime. We get the new words “seeven” and “seodd”. Numbers above seven must be odd, thodd, fodd, and seodd to be prime.

8 is even… 9 is theven… 10 is both even and feven…  None of these are therefore prime.

11 is the first number that is odd, thodd, fodd, and seodd so it become our next prime and we get the words “eleeven” and “eleodd”…

We can continue along this path forever. Perhaps our new terms theven and thodd might have some utility in the English language. I could say that my two friends and I only buy pizzas that are cut into a theven number of slices like 6 or 12 so we each fairly get an equal number of slices. If we buy a pizza cut into a thodd number of slices like 8, we are not able to distribute the slices fairly. But, it seems that seeven, seodd, eleeven and eleodd and further of these new word creations would have diminishing utility in every day life.

Now let’s return to the questions of why is 2 the only even prime number and what makes 2 special? Well, 2 is special because of the limitations of our current vocabulary. By now adding these new words to our vocabulary, we can see that “2 is the only even prime number”, “3 is the only theven prime number”, “5 is the only feven prime number”, “7 is the only seeven prime number”, "11 is the only eleeven prime number", and so on. 

Now, all the prime numbers are special in their own terms…