Is 2 Really the Oddest Prime Number?

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Discussion Overview

The discussion centers around the characterization of the prime number 2, particularly in relation to its status as the only even prime and the implications of this classification. Participants explore the significance of evenness in number theory, the uniqueness of 2 among primes, and the broader mathematical context of primes.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant argues that labeling 2 as the "oddest prime" is misleading, suggesting that being even is not inherently special compared to other divisibility properties.
  • Another participant counters that even numbers hold significant importance in group theory, implying a special status for 2.
  • Several participants emphasize the importance of distinguishing between odd and even integers, suggesting that this characteristic is more fundamental than other divisibility properties.
  • A participant notes that 2 uniquely eliminates half of the integers greater than itself from being prime, which they argue is a distinctive feature.
  • Another participant elaborates on the uniqueness of 2 by discussing how many general theorems about primes do not apply to it, particularly due to its relationship with the number 1.
  • One participant highlights that 2 is the first Prime Twin, suggesting this as another unique property of the number.

Areas of Agreement / Disagreement

Participants express differing views on the significance of 2 as an even prime and whether this characteristic is special compared to other properties of numbers. The discussion remains unresolved with multiple competing perspectives on the importance of evenness and the uniqueness of 2.

Contextual Notes

Some arguments hinge on the definitions of "special" and the relevance of various properties in number theory, which may not be universally agreed upon. The discussion also touches on the implications of theorems that do not apply to 2, indicating a nuanced understanding of its role among primes.

1MileCrash
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2 is the "oddest prime of all."

Regarding the old humorous "math joke" that 2 is the only even prime, thus it is the "oddest" prime of all. I have a bone to pick with this.

I don't think the idea of "even" numbers is any more special than numbers that are divisible by 3 or 5, or anything else. Divisibility by 2 just has a special name.

So we say that 2 is an 'odd' prime because it is the only prime that is even.
But, if we equivalently say that 2 is the only prime that is divisible by 2, we find that there is nothing special about this at all, because any prime p has the property that it is the only prime divisible by p by definition.

Thus, there is nothing odd about the prime number 2.

End rant.
 
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Well but even numbers are more important to group theory than numbers divisible by 3, 5, etc...
 
One of the most important characteristics of integers is whether they are odd or even. It's arguably much less important that a given integer has a remainder of 1 when divided by 3, or a remainder of 0 when divided by 6, for just two examples.
 
Mark44 said:
One of the most important characteristics of integers is whether they are odd or even. It's arguably much less important that a given integer has a remainder of 1 when divided by 3, or a remainder of 0 when divided by 6, for just two examples.

Why?
 
Thus, there is nothing odd about the prime number 2.

It as a single prime, takes out the largest amount of numbers from being prime.
50% of them greater than 2.

No other number unto itself can do that.
 
Last edited:
lostcauses10x said:
It as a single prime, takes out the largest amount of numbers from being prime.
50% of them greater than 2.

No other number unto itself can do that.

I don't know what you mean.
 
1MileCrash said:
I don't know what you mean.

I think he means that since 2 is the only even prime, all other even numbers (50% of ALL numbers greater than 2) are unable to be prime. That is rather unique.
 
Many general theorems about primes are not true for 2, and often the reason is that the proof for a prime p involves the number p-1. There is no question that 1 is a "special" number (because. 1x = x, for all values of x), and the general proof may break down when p-1 = 1, or it may need the fact that p-1 is even, or composite, for all primes except 2.

Actually the same thing is true for theorems about fields in general (fields being a class of mathematical objects which include real and complex numbers as two specific examples), which apply to all fields except the finite field with only two elements. That's why field theory has many theorems that start "If ... and ##1+1 \ne 0## then ..."
 
Well, 2 is the first Prime Twin (version Siamese) to appear.
That's a fairly unique property! :smile:
 

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