# What is an odd prime?

I heard the term "odd prime" often. Isn't it redundant? If $n$ is even then $2$ divides $n$ so it's not prime. Why is "odd" emphasized?

• ...and what about n=2? Mar 5, 2015 at 19:35
• I'm still looking for even primes greater than two... Mar 5, 2015 at 19:36
• Recall the definition of prime - a number which is divisible only by $1$ and by itself. So the fact that $2$ divides $n$ doesn't necessarily imply that $n$ is not prime. Mar 5, 2015 at 19:36
• While $\,2\,$ is a very odd prime, $\,{−}1\,$ is an even odder prime! $\ \$ Mar 5, 2015 at 20:06
• +1 this question sounds rather stupid, but is actually something that can confuse many if not properly understood. Mar 6, 2015 at 23:40

$2$ is an even prime, so "odd prime" is a short-hand for "a prime not equal to $2$".

• $2$ is the only even prime, making it an odd prime indeed. Mar 5, 2015 at 19:44
• @MartianInvader: How do you put 14 lumps of sugar into 3 teacups such that there is an odd number in each cup? One in the first, one in the second, and 12 in the last, because 12 is an odd number of lumps of sugar to put in one teacup, for certain. Mar 6, 2015 at 0:05
• This is an "odd" terminology methinks.
– AvZ
Mar 6, 2015 at 10:37
• I know it's standard, but I thoroughly dislike this terminology and always replace it by "a prime other than $2$". IMHO the terminology is utterly pretentious, in that it purports to exclude infinitely many cases, while in fact it only excludes one case. Mar 11, 2015 at 1:18

Ummm. I remember Michio Kuga saying “There are two types of primes, odd and even, and they cause the same amount of trouble.”

Irving Kaplansky was fond of quoting Marshall Hall about this: “It’s not that 2 is so small. It’s that it’s so even.”

From page xviii in T.Y.Lam, Introduction to Quadratic Forms over Fields:

A mathematician said “Who
Can quote me a theorem that's true?
For the ones that I know
Are simply not so
When the characteristic is two!”

• I don't buy this comment. It's because it is small, not because it is even. For instance, the cubic formula fails in characteristic 2 or 3. Because 2 is the smallest prime, it is a factor that affects more things than other primes. I see no reason that special cases involve 2 specifically because of its evenness. Oct 30, 2019 at 7:54

“All primes are odd, except $2$, which is the oddest of them all.”

(Graham, Knuth, Patashnik, Concrete Mathematics, second edition, page 129)

I swear I'm not yanking your chain on this one, but... $2$ is a prime number.

Human mathematicians used to define a prime number as an integer divisible by $1$ and by itself and no other positive integers. A little more than a hundred years ago, they decided to amend the definition to be an integer with exactly two positive divisors. This change excluded $1$ from the prime numbers, but $2$ still fits this new definition.

There are situations in which you need to focus on odd primes, such as for example, the computation of the Legendre symbol (see: http://mathworld.wolfram.com/LegendreSymbol.html, and be sure to click the "odd prime" link). There's a neat formula for the Legendre symbol. For $2$ you need the special Kronecker symbol (though I prefer to call it the Tooth Fairy symbol, as she came up with it centuries before Leopold Kronecker).

• Human mathematicians.. do other species have a different definition for primes? :) Mar 6, 2015 at 10:43
• @MikeMiller Avoiding anthropocentrism is a laudable goal. ;) Mar 6, 2015 at 17:14
• Humans are a special case of primates, which are the biological analogs of primes. Mar 6, 2015 at 20:36
• What I like about this answer is it provides an example where you would naturally see "odd prime". Mar 8, 2015 at 3:03
• @MikeMiller Maybe teh Vulcans distinguished between units and primes long before they developed cryptography, while the Klingons do subconsciously know their's a difference but just don't feel the need to articulate it. Pure speculation on my part, off course.
– user155234
Mar 10, 2015 at 21:09

While this might hug the the line of opinion, the literary focus of other answers provides some empirical support for the following argument:

The issue with even/odd primes isn't that 2 is prime, it's that we have the concept of oddness and evenness: The issue is in the language, not the math.

When we say that 2 is the only even prime, we mean:

2 is the only positive integer i such that i / 2 == 1 rem 0


'Even' is just a word that means "integer i such that i modulo 2 == 0". I argue that the number 2 in that definition isn't special; it just happens to be a function of our language that we have a word for it. Let us fix that with a new word: 'Threeven', which means "integer i such that i modulo 3 == 0", or in plain english, evenly divisible by three.

Shockingly, 3 is the only threeven prime, and we mean:

3 is the only positive integer i such that i / 3 == 1 rem 0


While threeven isn't (currently) part of the English language, perhaps a language formed by another species would include it if their biology intimated the importance of 3-divisibility: would they consider three to be an odder prime than 2? Perhaps of greater interest, will they similarly come up with awful puns about it?

In my eyes, 2 is a special prime not because of it's evenness, but because it is the first/smallest member in the sequence of natural primes.

(disclaimer: I'm not actually familiar with the Legendre Symbol, so I can't address that issue, and my math credentials have been rusting on the shelf for quite some time)

• I would suggest calling your new property "threeven". Mar 7, 2015 at 23:10
• @Floris: Nice. And done. Mar 8, 2015 at 3:40

A prime is an integer which has only two integer divisors that do not leave any remainder, one of those two integers is always one, the other is the integer itself.

The thing that some people find difficult is to recognise that the logic of prime numbers has nothing to do with whether a number is even or odd.

The confusion comes about because we have a special word for numbers that are divisible by 2 - we say they are 'even', and knowing that all even numbers are divisible by 2 some people assume that all even numbers are not primes. If we made up a word for numbers that are divisible by 3 - say 'TRIEN' we could make a similar sentence as the previous one and say: The confusion comes about because we have a special word for numbers that are divisible by 3 - we say they are 'trien', and knowing that all trien numbers are divisible by 3 some people assume that all trien numbers are not primes.

An 'odd prime' simply denotes all numbers that are prime and odd which includes all prime numbers except the number 2 which is both prime and even.

• I think the issue is just that it's easier to say "odd prime" than "prime greater than $2$". Mar 8, 2015 at 13:06
• I think it is more natural to define divisors and greatest common divisors and then coprimality as being a GCD of 1. Then the natural definition for what a prime is, is simply any integer which is coprime to all smaller integers. In this way, distinctness of factors and the arbitrary exclusion of 1 need not be brought up. Aug 9, 2020 at 0:41

Think of a Venn diagram consisting of two intersecting circles, one labeled ODD NUMBERS, the other labeled PRIMES. The number 1 is odd but not prime. The number 2 is prime but not odd. The intersection of the circles represents ODD PRIMES: 3, 5, 7,....