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The other prime numbers are all odd numbers such as $5, 11, 127,$ and $37$. So, why is $2$ the only prime even number there is?

Is it because it only has 1 and itself that way, even though it's even? Check it out on this excellent math page one-fourth from the bottom.

Theoretical Economist
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Mathster
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  • You mean even I guess... Do you know how prime numbers are defined? – Curious Droid Nov 03 '14 at 01:41
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    Just think, all even numbers are divisible by $2$, so they can't be prime. But if $2$ is itself, then it doesn't count. – Edward Jiang Nov 03 '14 at 01:42
  • A prime number has ${\large\tt\mbox{just two}}$ different divisors. So $1$ is not prime, $2$ is prime, etc... – Felix Marin Nov 03 '14 at 01:56
  • I don't know if it's possible to accept more than one answer and I don't want to tell you what to do but if you can sense that Lucian's answer also would have solved your problem even if user171177's answer hadn't existed, it may be worth considering putting a check mark beside that answer as well. I can't imagine having that question myself so it's hard to form my own opinion on what's a good answer. I already know the answer. Being even is equivalent to being a multiple of 2. 2 is a factor of every positive even number and 2 is the only even number that's small enough that 2 is equal to that – Timothy Mar 08 '20 at 01:24
  • number. Some people may have an intuition for something false that 2 should not have another special or unique property. For any prime number, you could take the property of what any positive integer is modulo that number. With 2, the fact that number is not a multiple of 2 gives away what it is modulo 2. On the other hand, the fact that a number is not a multiple of 3 doesn't give away what it is modulo 3. Sometimes an answer that gives the details of what's going on resolves the apparent contradiction. The answer – Timothy Mar 08 '20 at 01:39
  • https://math.stackexchange.com/questions/204632/the-concept-of-infinity/204639#204639 resolved the problem of the person who had a question about infinite sets. They probably had the intuition that adding another element to a set always gives you a larger set. If somebody asked a question about how it's possible that 2 has the unique property for a prime number that anything that's not a multiple of it is one more than a multiple of it, then one possible answer that may solve the author's problem is to explain how 2 is the only prime number that's so small that it is only 1 away from 1 and so – Timothy Mar 08 '20 at 01:48
  • it is the only prime number that is so small that all other prime numbers are one more than a multiple of that prime number. – Timothy Mar 08 '20 at 01:49
  • If $,|b|>1,$ then by definition $,ab$ is prime (i.e. irreducible) $!\iff! |a|=1.,$ OP is case $b=2.\ \ $ – Bill Dubuque Nov 08 '24 at 19:26

8 Answers8

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Why is two the only even $($binary$)$ number that is prime?

For the same reason that three is the only ternary number that is prime. Which is the same reason for which five is the only quinary number that is prime. Etc.

Lucian
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    This is the best answer! – HEKTO Nov 03 '14 at 03:09
  • I already had an answer like this and read the answer by user171177 first. I already saw that answer as including this answer in it so I upvoted it and saw how it answered the question. Maybe if everyone was as good as me at it, that answer would have gotten a score of 22. For me, this answer doesn't really add anything to it. That answer is performing at 100% already. – Timothy Sep 10 '21 at 03:51
  • @Timothy: Same message, different package. ;-) – Lucian Sep 10 '21 at 05:25
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Because every even number other than 2 is obviously divisible by 2 and so by definition cannot be prime.

John Kontol
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  • That makes it sound like 2 is not divisible by 2. But it is. All even numbers are divisible by 2. – Ocean Dec 10 '24 at 17:52
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A (positive) even number is some number $n$ such that $n = 2 \cdot k$ for some (positive) integer $k$. A prime number has only itself and $1$ as (positive) divisors.

What happens if $n \not = 2$ in our definition of even numbers?

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Pick a prime $p$. Call a number $n$ $p$-divisible if $p\mid n$. Then $p$ is the only $p$-divisible prime, trivially. In particular, $2$ is the only $2$-divisible, or even, prime.

Pedro
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In the integers, $-2$ is another even prime.

For variety, in the Gaussian integers, $2$ is not prime: e.g. factors as $(1+i)(1-i)$. The even primes of the Gaussian integers are $\pm 1 \pm i$, although these are all the "same" prime in the same sense that in the integers, $2$ and $-2$ are the "same" prime.

(I define "even" in a number field to be equivalent to its norm being even)

In the ring of all rational numbers with odd denominator, $2/7$ is an even prime. In fact, $2/n$ is prime for every odd integer $n$. (but again, these are all the "same" prime)

There are also number rings that have distinct even primes that are not the "same" in the sense implied above.

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A prime number is such that it is divisible by only itself and one. Including 1 as a prime number would violate the fundamental theory of arithmetic, so in modern mathematics it is excluded. Two is a prime because it is divisible by only two and one. All the other even numbers are not prime because they are all divisible by two. That leaves only the odd numbers. Of course, not all odd numbers are prime (e.g. nine is divisible by three).

Michael Lee
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The word prime comes from the Latin word primus which means "first." Two (2) is the first even number. In other words, it starts all the even numbers. There is more than one odd prime number because odd numbers are never divisible by 2.

Carly Brooke Steffen
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  • This explanation of the word origin is doubtful. Prime numbers are called prime because they are the "building blocks", the "atoms" of whole numbers. In contrast to /composite/ numbers. – mathematician Mar 26 '17 at 00:41
  • That's why they are considered what would come first as of numbers. A composite number gets its name because it's composed of at least one prime number. – Carly Brooke Steffen Mar 27 '17 at 03:03
  • @mathematician That is not true. They are called "prime" because they are "the first in the list". Take a list of all positive integers. Starting with $2$, circle all numbers that have not been circled yet, and cross out all numbers that are multiples of previous numbers. What you get are the primes, i.e., the first numbers in the list that are not multiples of other numbers. – Klangen Sep 19 '18 at 22:15
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The number $2$ has only two whole number factors, $1$ and itself. That's pretty much it after this: The other numbers that are even up from two are all divisible by that number in some way. This is also known as the "oddest prime" because it's the only prime number that's even, so it's also known as the odd one out. I guess now that that's pretty much it going to the question about why this can happen.

Mathster
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