Why is the definition of prime numbers written to include 2
$begingroup$
This question is often asked as "why is 2 a prime number" and the only answers I can find are "The definition of prime numbers is written in such a way that 2 is prime". Sometimes, if the question was "Why is 2 a prime number even though it's even" the answer will include some explanation that being even is not such a special property, it just means a number is divisible by two and of course two is divisible by two, that's the number itself and three is divisible by three and it's still prime and so on.
But there is something special about a number being even, and it 2 does have a special property that is not shared by any other prime number. It can be divided into equal subsets.
The answer I'm looking for will answer the question in the title and also answer: If we change the definition of a prime number, what effects does that have on mathematics theories, proofs, etc. Are certain claims that were useful and important under the old definition no longer true?
Specifically, What's the impact of changing the definition of prime to this: A prime is any integer that cannot be divided into smaller equal groups. N.B. this also changes the primeness of 1 but not any other number.
prime-numbers definition
asked 2 hours ago
SegfaultSegfault
1093
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|
show 12 more comments
$begingroup$
This question is often asked as "why is 2 a prime number" and the only answers I can find are "The definition of prime numbers is written in such a way that 2 is prime". Sometimes, if the question was "Why is 2 a prime number even though it's even" the answer will include some explanation that being even is not such a special property, it just means a number is divisible by two and of course two is divisible by two, that's the number itself and three is divisible by three and it's still prime and so on.
But there is something special about a number being even, and it 2 does have a special property that is not shared by any other prime number. It can be divided into equal subsets.
The answer I'm looking for will answer the question in the title and also answer: If we change the definition of a prime number, what effects does that have on mathematics theories, proofs, etc. Are certain claims that were useful and important under the old definition no longer true?
Specifically, What's the impact of changing the definition of prime to this: A prime is any integer that cannot be divided into smaller equal groups. N.B. this also changes the primeness of 1 but not any other number.
prime-numbers definition
asked 2 hours ago
SegfaultSegfault
1093
New contributor
Segfault is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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4
$begingroup$
It would mess up the theory of prime factorisation if $2$ weren't prime....
$endgroup$
– Lord Shark the Unknown
2 hours ago
8
$begingroup$
Why is $3$ a prime number? It's the only prime divisible by $3$. Shouldn't that be special?
$endgroup$
– saulspatz
2 hours ago
3
$begingroup$
A number $n$ is prime when has exactly two divisors : $1$ and $n$ itself. See Euclid's Elements, BK. VII, Def.11 : "A prime number is that which is measured by a unit alone."
$endgroup$
– Mauro ALLEGRANZA
2 hours ago
6
$begingroup$
There is nothing special about the ability to be divided into smaller equal parts. Every natural number $n$ can be divided into $n$ groups of $1$.
$endgroup$
– Lee Mosher
2 hours ago
2
$begingroup$
It can be divided into equal subsets. --- I realize that you might have meant "divided" as split into half (like a sword slicing through the set only once, producing equal size parts), but in this case you're simply talking about integers divisible by $2$ and thus defining what you want to exclude in terms of itself (i.e. the argument carries little to no persuasive or rhetorical weight). I would have included this in my other comment, but ran afoul of the comment character limitations. By the way, I don't think the question is all that bad if you're just beginning to learn about primes.
$endgroup$
– Dave L. Renfro
2 hours ago
|
show 12 more comments
$begingroup$
This question is often asked as "why is 2 a prime number" and the only answers I can find are "The definition of prime numbers is written in such a way that 2 is prime". Sometimes, if the question was "Why is 2 a prime number even though it's even" the answer will include some explanation that being even is not such a special property, it just means a number is divisible by two and of course two is divisible by two, that's the number itself and three is divisible by three and it's still prime and so on.
But there is something special about a number being even, and it 2 does have a special property that is not shared by any other prime number. It can be divided into equal subsets.
The answer I'm looking for will answer the question in the title and also answer: If we change the definition of a prime number, what effects does that have on mathematics theories, proofs, etc. Are certain claims that were useful and important under the old definition no longer true?
Specifically, What's the impact of changing the definition of prime to this: A prime is any integer that cannot be divided into smaller equal groups. N.B. this also changes the primeness of 1 but not any other number.
prime-numbers definition
asked 2 hours ago
SegfaultSegfault
1093
New contributor
Segfault is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
This question is often asked as "why is 2 a prime number" and the only answers I can find are "The definition of prime numbers is written in such a way that 2 is prime". Sometimes, if the question was "Why is 2 a prime number even though it's even" the answer will include some explanation that being even is not such a special property, it just means a number is divisible by two and of course two is divisible by two, that's the number itself and three is divisible by three and it's still prime and so on.
But there is something special about a number being even, and it 2 does have a special property that is not shared by any other prime number. It can be divided into equal subsets.
The answer I'm looking for will answer the question in the title and also answer: If we change the definition of a prime number, what effects does that have on mathematics theories, proofs, etc. Are certain claims that were useful and important under the old definition no longer true?
Specifically, What's the impact of changing the definition of prime to this: A prime is any integer that cannot be divided into smaller equal groups. N.B. this also changes the primeness of 1 but not any other number.
prime-numbers definition
prime-numbers definition
asked 2 hours ago
SegfaultSegfault
1093
New contributor
Segfault is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
SegfaultSegfault
1093
New contributor
Segfault is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
SegfaultSegfault
1093
New contributor
Segfault is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
SegfaultSegfault
1093
asked 2 hours ago
SegfaultSegfault
1093
1093
New contributor
Segfault is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
Segfault is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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4
$begingroup$
It would mess up the theory of prime factorisation if $2$ weren't prime....
$endgroup$
– Lord Shark the Unknown
2 hours ago
8
$begingroup$
Why is $3$ a prime number? It's the only prime divisible by $3$. Shouldn't that be special?
$endgroup$
– saulspatz
2 hours ago
3
$begingroup$
A number $n$ is prime when has exactly two divisors : $1$ and $n$ itself. See Euclid's Elements, BK. VII, Def.11 : "A prime number is that which is measured by a unit alone."
$endgroup$
– Mauro ALLEGRANZA
2 hours ago
6
$begingroup$
There is nothing special about the ability to be divided into smaller equal parts. Every natural number $n$ can be divided into $n$ groups of $1$.
$endgroup$
– Lee Mosher
2 hours ago
2
$begingroup$
It can be divided into equal subsets. --- I realize that you might have meant "divided" as split into half (like a sword slicing through the set only once, producing equal size parts), but in this case you're simply talking about integers divisible by $2$ and thus defining what you want to exclude in terms of itself (i.e. the argument carries little to no persuasive or rhetorical weight). I would have included this in my other comment, but ran afoul of the comment character limitations. By the way, I don't think the question is all that bad if you're just beginning to learn about primes.
$endgroup$
– Dave L. Renfro
2 hours ago
|
show 12 more comments
4
$begingroup$
It would mess up the theory of prime factorisation if $2$ weren't prime....
$endgroup$
– Lord Shark the Unknown
2 hours ago
8
$begingroup$
Why is $3$ a prime number? It's the only prime divisible by $3$. Shouldn't that be special?
$endgroup$
– saulspatz
2 hours ago
3
$begingroup$
A number $n$ is prime when has exactly two divisors : $1$ and $n$ itself. See Euclid's Elements, BK. VII, Def.11 : "A prime number is that which is measured by a unit alone."
$endgroup$
– Mauro ALLEGRANZA
2 hours ago
6
$begingroup$
There is nothing special about the ability to be divided into smaller equal parts. Every natural number $n$ can be divided into $n$ groups of $1$.
$endgroup$
– Lee Mosher
2 hours ago
2
$begingroup$
It can be divided into equal subsets. --- I realize that you might have meant "divided" as split into half (like a sword slicing through the set only once, producing equal size parts), but in this case you're simply talking about integers divisible by $2$ and thus defining what you want to exclude in terms of itself (i.e. the argument carries little to no persuasive or rhetorical weight). I would have included this in my other comment, but ran afoul of the comment character limitations. By the way, I don't think the question is all that bad if you're just beginning to learn about primes.
$endgroup$
– Dave L. Renfro
2 hours ago
4
4
$begingroup$
It would mess up the theory of prime factorisation if $2$ weren't prime....
$endgroup$
– Lord Shark the Unknown
2 hours ago
$begingroup$
It would mess up the theory of prime factorisation if $2$ weren't prime....
$endgroup$
– Lord Shark the Unknown
2 hours ago
8
8
$begingroup$
Why is $3$ a prime number? It's the only prime divisible by $3$. Shouldn't that be special?
$endgroup$
– saulspatz
2 hours ago
$begingroup$
Why is $3$ a prime number? It's the only prime divisible by $3$. Shouldn't that be special?
$endgroup$
– saulspatz
2 hours ago
3
3
$begingroup$
A number $n$ is prime when has exactly two divisors : $1$ and $n$ itself. See Euclid's Elements, BK. VII, Def.11 : "A prime number is that which is measured by a unit alone."
$endgroup$
– Mauro ALLEGRANZA
2 hours ago
$begingroup$
A number $n$ is prime when has exactly two divisors : $1$ and $n$ itself. See Euclid's Elements, BK. VII, Def.11 : "A prime number is that which is measured by a unit alone."
$endgroup$
– Mauro ALLEGRANZA
2 hours ago
6
6
$begingroup$
There is nothing special about the ability to be divided into smaller equal parts. Every natural number $n$ can be divided into $n$ groups of $1$.
$endgroup$
– Lee Mosher
2 hours ago
$begingroup$
There is nothing special about the ability to be divided into smaller equal parts. Every natural number $n$ can be divided into $n$ groups of $1$.
$endgroup$
– Lee Mosher
2 hours ago
2
2
$begingroup$
It can be divided into equal subsets. --- I realize that you might have meant "divided" as split into half (like a sword slicing through the set only once, producing equal size parts), but in this case you're simply talking about integers divisible by $2$ and thus defining what you want to exclude in terms of itself (i.e. the argument carries little to no persuasive or rhetorical weight). I would have included this in my other comment, but ran afoul of the comment character limitations. By the way, I don't think the question is all that bad if you're just beginning to learn about primes.
$endgroup$
– Dave L. Renfro
2 hours ago
$begingroup$
It can be divided into equal subsets. --- I realize that you might have meant "divided" as split into half (like a sword slicing through the set only once, producing equal size parts), but in this case you're simply talking about integers divisible by $2$ and thus defining what you want to exclude in terms of itself (i.e. the argument carries little to no persuasive or rhetorical weight). I would have included this in my other comment, but ran afoul of the comment character limitations. By the way, I don't think the question is all that bad if you're just beginning to learn about primes.
$endgroup$
– Dave L. Renfro
2 hours ago
|
show 12 more comments
3 Answers
3
active
oldest
votes
$begingroup$
We define things in such a way that they have some practical use, and help us make sense of the world around us. This is exactly how words and concepts are created, and also evolve. For example, we decided to give a name to a class of objects in the sky with similar behavior: 'planets'. These objects form what a philosopher might call a 'natural' class of objects. Having labels for them allows us to talk and think about those objects more easily, helping us with explanations, predictions, doing science, and again making sense of things in general.
But like I said, definitions can evolve: Pluto is no longer considered a 'planet', because after finding out more about our solar system we realized Pluto is in significant ways different from Neptune, Jupiter, Earth, etc. That is, by putting Pluto into different (though still related) class of 'dwarf-planets', we now look at it a little differently.
The same holds for mathematical definitions. For example, we could define 'huppelflup numbers' to be exactly those numbers that can be divided by 17 or by 631 ... but there just isn't much practical use to such a definition, and so we don't.
But the way we define prime numbers has lots of practical uses. For example, with the current definition, we get the nice, clean, result that every number has a unique prime factorization. And it's not just applications within mathematics that matter: prime numbers have tremendous importance for real life as well.
Now, if we were to exclude $2$ as a prime, this would no longer be true. And a bunch of other results would likewise have to be stated in a much more cumbersome way.
And by the way, this unique prime factorization theorem is exactly why mathematicians did exclude $1$ as a prime .. even though originally it was.
So yes, you're right that it is not as if definitions are fixed until the end of time. And maybe at some point in the future we redefine the sets of primes again to also exclude $2$, because doing so will have some other advantages.
However, I wouldn't hold my breadth: the current definition is very nice.
answered 2 hours ago
Bram28Bram28
60.9k44590
$endgroup$
$begingroup$
Thank you, that's exactly what I was wondering
$endgroup$
– Segfault
2 hours ago
1
$begingroup$
@Segfault I figured ... It's good you ask these kinds of questions, and hold our feet to the fire. For someone to say: 'but that's just what the definition is!" is really not enough; there are reasons to be made for defining things a certain way in the first place, and sometimes we may want to revisit those reasons and see if they are still most useful.
$endgroup$
– Bram28
2 hours ago
add a comment |
$begingroup$
$2$ is considered a prime because most of the time that turns out to be convenient. $1$ is not considered a prime for the same reason.
That being said, $2$ is definitely the oddest prime.
answered 2 hours ago
ArthurArthur
113k7110193
$endgroup$
$begingroup$
Sorry, I don't understand why it's convenient for two to be a prime.
$endgroup$
– Segfault
2 hours ago
2
$begingroup$
@Segfault Almost every single theorem concerning primes or non-primes is a reason why it's convenient. Not a single one of them can be pointed to as the reason (although the fundamental theorem of arithmetic is a strong contender), but together they make a compelling argument.
$endgroup$
– Arthur
2 hours ago
2
$begingroup$
So the oddest prime is even! :o)
$endgroup$
– Bernard
2 hours ago
add a comment |
$begingroup$
You may wish to read the 2012 Journal of Integer Sequences article What is the Smallest Prime?, by Chris K. Caldwell and Yeng Xiong. The abstract starts with
What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers.
Also, as already basically stated in the comments and other answers, the Introduction says
... whether or not a number (especially unity) is a prime is a matter of definition, so a matter of choice, context and tradition, not a matter of proof. Yet definitions are not made at random; these choices are bound by our usage of mathematics and, especially in this case, by our notation.
I enjoyed reading this article & found it educational. The only thing I have to add to this article & what's already been stated here is that I also believe it's generally good to question things instead of just accepting the status quo because "that's the way it is". During my years tutoring math at university, I had a philosophy of "there's no such thing as a stupid question, only a stupid answer". What I mean is that if the person has made a reasonable effort to resolve it on their own and the question is sincere, it's deserving of a reasonable answer.
answered 44 mins ago
John OmielanJohn Omielan
1,73629
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add a comment |
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3 Answers
3
active
oldest
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3 Answers
3
active
oldest
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active
oldest
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active
oldest
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$begingroup$
We define things in such a way that they have some practical use, and help us make sense of the world around us. This is exactly how words and concepts are created, and also evolve. For example, we decided to give a name to a class of objects in the sky with similar behavior: 'planets'. These objects form what a philosopher might call a 'natural' class of objects. Having labels for them allows us to talk and think about those objects more easily, helping us with explanations, predictions, doing science, and again making sense of things in general.
But like I said, definitions can evolve: Pluto is no longer considered a 'planet', because after finding out more about our solar system we realized Pluto is in significant ways different from Neptune, Jupiter, Earth, etc. That is, by putting Pluto into different (though still related) class of 'dwarf-planets', we now look at it a little differently.
The same holds for mathematical definitions. For example, we could define 'huppelflup numbers' to be exactly those numbers that can be divided by 17 or by 631 ... but there just isn't much practical use to such a definition, and so we don't.
But the way we define prime numbers has lots of practical uses. For example, with the current definition, we get the nice, clean, result that every number has a unique prime factorization. And it's not just applications within mathematics that matter: prime numbers have tremendous importance for real life as well.
Now, if we were to exclude $2$ as a prime, this would no longer be true. And a bunch of other results would likewise have to be stated in a much more cumbersome way.
And by the way, this unique prime factorization theorem is exactly why mathematicians did exclude $1$ as a prime .. even though originally it was.
So yes, you're right that it is not as if definitions are fixed until the end of time. And maybe at some point in the future we redefine the sets of primes again to also exclude $2$, because doing so will have some other advantages.
However, I wouldn't hold my breadth: the current definition is very nice.
answered 2 hours ago
Bram28Bram28
60.9k44590
$endgroup$
$begingroup$
Thank you, that's exactly what I was wondering
$endgroup$
– Segfault
2 hours ago
1
$begingroup$
@Segfault I figured ... It's good you ask these kinds of questions, and hold our feet to the fire. For someone to say: 'but that's just what the definition is!" is really not enough; there are reasons to be made for defining things a certain way in the first place, and sometimes we may want to revisit those reasons and see if they are still most useful.
$endgroup$
– Bram28
2 hours ago
add a comment |
$begingroup$
We define things in such a way that they have some practical use, and help us make sense of the world around us. This is exactly how words and concepts are created, and also evolve. For example, we decided to give a name to a class of objects in the sky with similar behavior: 'planets'. These objects form what a philosopher might call a 'natural' class of objects. Having labels for them allows us to talk and think about those objects more easily, helping us with explanations, predictions, doing science, and again making sense of things in general.
But like I said, definitions can evolve: Pluto is no longer considered a 'planet', because after finding out more about our solar system we realized Pluto is in significant ways different from Neptune, Jupiter, Earth, etc. That is, by putting Pluto into different (though still related) class of 'dwarf-planets', we now look at it a little differently.
The same holds for mathematical definitions. For example, we could define 'huppelflup numbers' to be exactly those numbers that can be divided by 17 or by 631 ... but there just isn't much practical use to such a definition, and so we don't.
But the way we define prime numbers has lots of practical uses. For example, with the current definition, we get the nice, clean, result that every number has a unique prime factorization. And it's not just applications within mathematics that matter: prime numbers have tremendous importance for real life as well.
Now, if we were to exclude $2$ as a prime, this would no longer be true. And a bunch of other results would likewise have to be stated in a much more cumbersome way.
And by the way, this unique prime factorization theorem is exactly why mathematicians did exclude $1$ as a prime .. even though originally it was.
So yes, you're right that it is not as if definitions are fixed until the end of time. And maybe at some point in the future we redefine the sets of primes again to also exclude $2$, because doing so will have some other advantages.
However, I wouldn't hold my breadth: the current definition is very nice.
answered 2 hours ago
Bram28Bram28
60.9k44590
$endgroup$
$begingroup$
Thank you, that's exactly what I was wondering
$endgroup$
– Segfault
2 hours ago
1
$begingroup$
@Segfault I figured ... It's good you ask these kinds of questions, and hold our feet to the fire. For someone to say: 'but that's just what the definition is!" is really not enough; there are reasons to be made for defining things a certain way in the first place, and sometimes we may want to revisit those reasons and see if they are still most useful.
$endgroup$
– Bram28
2 hours ago
add a comment |
$begingroup$
We define things in such a way that they have some practical use, and help us make sense of the world around us. This is exactly how words and concepts are created, and also evolve. For example, we decided to give a name to a class of objects in the sky with similar behavior: 'planets'. These objects form what a philosopher might call a 'natural' class of objects. Having labels for them allows us to talk and think about those objects more easily, helping us with explanations, predictions, doing science, and again making sense of things in general.
But like I said, definitions can evolve: Pluto is no longer considered a 'planet', because after finding out more about our solar system we realized Pluto is in significant ways different from Neptune, Jupiter, Earth, etc. That is, by putting Pluto into different (though still related) class of 'dwarf-planets', we now look at it a little differently.
The same holds for mathematical definitions. For example, we could define 'huppelflup numbers' to be exactly those numbers that can be divided by 17 or by 631 ... but there just isn't much practical use to such a definition, and so we don't.
But the way we define prime numbers has lots of practical uses. For example, with the current definition, we get the nice, clean, result that every number has a unique prime factorization. And it's not just applications within mathematics that matter: prime numbers have tremendous importance for real life as well.
Now, if we were to exclude $2$ as a prime, this would no longer be true. And a bunch of other results would likewise have to be stated in a much more cumbersome way.
And by the way, this unique prime factorization theorem is exactly why mathematicians did exclude $1$ as a prime .. even though originally it was.
So yes, you're right that it is not as if definitions are fixed until the end of time. And maybe at some point in the future we redefine the sets of primes again to also exclude $2$, because doing so will have some other advantages.
However, I wouldn't hold my breadth: the current definition is very nice.
answered 2 hours ago
Bram28Bram28
60.9k44590
$endgroup$
We define things in such a way that they have some practical use, and help us make sense of the world around us. This is exactly how words and concepts are created, and also evolve. For example, we decided to give a name to a class of objects in the sky with similar behavior: 'planets'. These objects form what a philosopher might call a 'natural' class of objects. Having labels for them allows us to talk and think about those objects more easily, helping us with explanations, predictions, doing science, and again making sense of things in general.
But like I said, definitions can evolve: Pluto is no longer considered a 'planet', because after finding out more about our solar system we realized Pluto is in significant ways different from Neptune, Jupiter, Earth, etc. That is, by putting Pluto into different (though still related) class of 'dwarf-planets', we now look at it a little differently.
The same holds for mathematical definitions. For example, we could define 'huppelflup numbers' to be exactly those numbers that can be divided by 17 or by 631 ... but there just isn't much practical use to such a definition, and so we don't.
But the way we define prime numbers has lots of practical uses. For example, with the current definition, we get the nice, clean, result that every number has a unique prime factorization. And it's not just applications within mathematics that matter: prime numbers have tremendous importance for real life as well.
Now, if we were to exclude $2$ as a prime, this would no longer be true. And a bunch of other results would likewise have to be stated in a much more cumbersome way.
And by the way, this unique prime factorization theorem is exactly why mathematicians did exclude $1$ as a prime .. even though originally it was.
So yes, you're right that it is not as if definitions are fixed until the end of time. And maybe at some point in the future we redefine the sets of primes again to also exclude $2$, because doing so will have some other advantages.
However, I wouldn't hold my breadth: the current definition is very nice.
answered 2 hours ago
Bram28Bram28
60.9k44590
edited 2 hours ago
answered 2 hours ago
Bram28Bram28
60.9k44590
answered 2 hours ago
Bram28Bram28
60.9k44590
answered 2 hours ago
Bram28Bram28
60.9k44590
60.9k44590
$begingroup$
Thank you, that's exactly what I was wondering
$endgroup$
– Segfault
2 hours ago
1
$begingroup$
@Segfault I figured ... It's good you ask these kinds of questions, and hold our feet to the fire. For someone to say: 'but that's just what the definition is!" is really not enough; there are reasons to be made for defining things a certain way in the first place, and sometimes we may want to revisit those reasons and see if they are still most useful.
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– Bram28
2 hours ago
add a comment |
$begingroup$
Thank you, that's exactly what I was wondering
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– Segfault
2 hours ago
1
$begingroup$
@Segfault I figured ... It's good you ask these kinds of questions, and hold our feet to the fire. For someone to say: 'but that's just what the definition is!" is really not enough; there are reasons to be made for defining things a certain way in the first place, and sometimes we may want to revisit those reasons and see if they are still most useful.
$endgroup$
– Bram28
2 hours ago
$begingroup$
Thank you, that's exactly what I was wondering
$endgroup$
– Segfault
2 hours ago
$begingroup$
Thank you, that's exactly what I was wondering
$endgroup$
– Segfault
2 hours ago
1
1
$begingroup$
@Segfault I figured ... It's good you ask these kinds of questions, and hold our feet to the fire. For someone to say: 'but that's just what the definition is!" is really not enough; there are reasons to be made for defining things a certain way in the first place, and sometimes we may want to revisit those reasons and see if they are still most useful.
$endgroup$
– Bram28
2 hours ago
$begingroup$
@Segfault I figured ... It's good you ask these kinds of questions, and hold our feet to the fire. For someone to say: 'but that's just what the definition is!" is really not enough; there are reasons to be made for defining things a certain way in the first place, and sometimes we may want to revisit those reasons and see if they are still most useful.
$endgroup$
– Bram28
2 hours ago
add a comment |
$begingroup$
$2$ is considered a prime because most of the time that turns out to be convenient. $1$ is not considered a prime for the same reason.
That being said, $2$ is definitely the oddest prime.
answered 2 hours ago
ArthurArthur
113k7110193
$endgroup$
$begingroup$
Sorry, I don't understand why it's convenient for two to be a prime.
$endgroup$
– Segfault
2 hours ago
2
$begingroup$
@Segfault Almost every single theorem concerning primes or non-primes is a reason why it's convenient. Not a single one of them can be pointed to as the reason (although the fundamental theorem of arithmetic is a strong contender), but together they make a compelling argument.
$endgroup$
– Arthur
2 hours ago
2
$begingroup$
So the oddest prime is even! :o)
$endgroup$
– Bernard
2 hours ago
add a comment |
$begingroup$
$2$ is considered a prime because most of the time that turns out to be convenient. $1$ is not considered a prime for the same reason.
That being said, $2$ is definitely the oddest prime.
answered 2 hours ago
ArthurArthur
113k7110193
$endgroup$
$begingroup$
Sorry, I don't understand why it's convenient for two to be a prime.
$endgroup$
– Segfault
2 hours ago
2
$begingroup$
@Segfault Almost every single theorem concerning primes or non-primes is a reason why it's convenient. Not a single one of them can be pointed to as the reason (although the fundamental theorem of arithmetic is a strong contender), but together they make a compelling argument.
$endgroup$
– Arthur
2 hours ago
2
$begingroup$
So the oddest prime is even! :o)
$endgroup$
– Bernard
2 hours ago
add a comment |
$begingroup$
$2$ is considered a prime because most of the time that turns out to be convenient. $1$ is not considered a prime for the same reason.
That being said, $2$ is definitely the oddest prime.
answered 2 hours ago
ArthurArthur
113k7110193
$endgroup$
$2$ is considered a prime because most of the time that turns out to be convenient. $1$ is not considered a prime for the same reason.
That being said, $2$ is definitely the oddest prime.
answered 2 hours ago
ArthurArthur
113k7110193
answered 2 hours ago
ArthurArthur
113k7110193
answered 2 hours ago
ArthurArthur
113k7110193
answered 2 hours ago
ArthurArthur
113k7110193
113k7110193
$begingroup$
Sorry, I don't understand why it's convenient for two to be a prime.
$endgroup$
– Segfault
2 hours ago
2
$begingroup$
@Segfault Almost every single theorem concerning primes or non-primes is a reason why it's convenient. Not a single one of them can be pointed to as the reason (although the fundamental theorem of arithmetic is a strong contender), but together they make a compelling argument.
$endgroup$
– Arthur
2 hours ago
2
$begingroup$
So the oddest prime is even! :o)
$endgroup$
– Bernard
2 hours ago
add a comment |
$begingroup$
Sorry, I don't understand why it's convenient for two to be a prime.
$endgroup$
– Segfault
2 hours ago
2
$begingroup$
@Segfault Almost every single theorem concerning primes or non-primes is a reason why it's convenient. Not a single one of them can be pointed to as the reason (although the fundamental theorem of arithmetic is a strong contender), but together they make a compelling argument.
$endgroup$
– Arthur
2 hours ago
2
$begingroup$
So the oddest prime is even! :o)
$endgroup$
– Bernard
2 hours ago
$begingroup$
Sorry, I don't understand why it's convenient for two to be a prime.
$endgroup$
– Segfault
2 hours ago
$begingroup$
Sorry, I don't understand why it's convenient for two to be a prime.
$endgroup$
– Segfault
2 hours ago
2
2
$begingroup$
@Segfault Almost every single theorem concerning primes or non-primes is a reason why it's convenient. Not a single one of them can be pointed to as the reason (although the fundamental theorem of arithmetic is a strong contender), but together they make a compelling argument.
$endgroup$
– Arthur
2 hours ago
$begingroup$
@Segfault Almost every single theorem concerning primes or non-primes is a reason why it's convenient. Not a single one of them can be pointed to as the reason (although the fundamental theorem of arithmetic is a strong contender), but together they make a compelling argument.
$endgroup$
– Arthur
2 hours ago
2
2
$begingroup$
So the oddest prime is even! :o)
$endgroup$
– Bernard
2 hours ago
$begingroup$
So the oddest prime is even! :o)
$endgroup$
– Bernard
2 hours ago
add a comment |
$begingroup$
You may wish to read the 2012 Journal of Integer Sequences article What is the Smallest Prime?, by Chris K. Caldwell and Yeng Xiong. The abstract starts with
What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers.
Also, as already basically stated in the comments and other answers, the Introduction says
... whether or not a number (especially unity) is a prime is a matter of definition, so a matter of choice, context and tradition, not a matter of proof. Yet definitions are not made at random; these choices are bound by our usage of mathematics and, especially in this case, by our notation.
I enjoyed reading this article & found it educational. The only thing I have to add to this article & what's already been stated here is that I also believe it's generally good to question things instead of just accepting the status quo because "that's the way it is". During my years tutoring math at university, I had a philosophy of "there's no such thing as a stupid question, only a stupid answer". What I mean is that if the person has made a reasonable effort to resolve it on their own and the question is sincere, it's deserving of a reasonable answer.
answered 44 mins ago
John OmielanJohn Omielan
1,73629
$endgroup$
add a comment |
$begingroup$
You may wish to read the 2012 Journal of Integer Sequences article What is the Smallest Prime?, by Chris K. Caldwell and Yeng Xiong. The abstract starts with
What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers.
Also, as already basically stated in the comments and other answers, the Introduction says
... whether or not a number (especially unity) is a prime is a matter of definition, so a matter of choice, context and tradition, not a matter of proof. Yet definitions are not made at random; these choices are bound by our usage of mathematics and, especially in this case, by our notation.
I enjoyed reading this article & found it educational. The only thing I have to add to this article & what's already been stated here is that I also believe it's generally good to question things instead of just accepting the status quo because "that's the way it is". During my years tutoring math at university, I had a philosophy of "there's no such thing as a stupid question, only a stupid answer". What I mean is that if the person has made a reasonable effort to resolve it on their own and the question is sincere, it's deserving of a reasonable answer.
answered 44 mins ago
John OmielanJohn Omielan
1,73629
$endgroup$
add a comment |
$begingroup$
You may wish to read the 2012 Journal of Integer Sequences article What is the Smallest Prime?, by Chris K. Caldwell and Yeng Xiong. The abstract starts with
What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers.
Also, as already basically stated in the comments and other answers, the Introduction says
... whether or not a number (especially unity) is a prime is a matter of definition, so a matter of choice, context and tradition, not a matter of proof. Yet definitions are not made at random; these choices are bound by our usage of mathematics and, especially in this case, by our notation.
I enjoyed reading this article & found it educational. The only thing I have to add to this article & what's already been stated here is that I also believe it's generally good to question things instead of just accepting the status quo because "that's the way it is". During my years tutoring math at university, I had a philosophy of "there's no such thing as a stupid question, only a stupid answer". What I mean is that if the person has made a reasonable effort to resolve it on their own and the question is sincere, it's deserving of a reasonable answer.
answered 44 mins ago
John OmielanJohn Omielan
1,73629
$endgroup$
You may wish to read the 2012 Journal of Integer Sequences article What is the Smallest Prime?, by Chris K. Caldwell and Yeng Xiong. The abstract starts with
What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers.
Also, as already basically stated in the comments and other answers, the Introduction says
... whether or not a number (especially unity) is a prime is a matter of definition, so a matter of choice, context and tradition, not a matter of proof. Yet definitions are not made at random; these choices are bound by our usage of mathematics and, especially in this case, by our notation.
I enjoyed reading this article & found it educational. The only thing I have to add to this article & what's already been stated here is that I also believe it's generally good to question things instead of just accepting the status quo because "that's the way it is". During my years tutoring math at university, I had a philosophy of "there's no such thing as a stupid question, only a stupid answer". What I mean is that if the person has made a reasonable effort to resolve it on their own and the question is sincere, it's deserving of a reasonable answer.
answered 44 mins ago
John OmielanJohn Omielan
1,73629
edited 37 mins ago
answered 44 mins ago
John OmielanJohn Omielan
1,73629
answered 44 mins ago
John OmielanJohn Omielan
1,73629
answered 44 mins ago
John OmielanJohn Omielan
1,73629
1,73629
add a comment |
add a comment |
Segfault is a new contributor. Be nice, and check out our Code of Conduct.
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4
$begingroup$
It would mess up the theory of prime factorisation if $2$ weren't prime....
$endgroup$
– Lord Shark the Unknown
2 hours ago
8
$begingroup$
Why is $3$ a prime number? It's the only prime divisible by $3$. Shouldn't that be special?
$endgroup$
– saulspatz
2 hours ago
3
$begingroup$
A number $n$ is prime when has exactly two divisors : $1$ and $n$ itself. See Euclid's Elements, BK. VII, Def.11 : "A prime number is that which is measured by a unit alone."
$endgroup$
– Mauro ALLEGRANZA
2 hours ago
6
$begingroup$
There is nothing special about the ability to be divided into smaller equal parts. Every natural number $n$ can be divided into $n$ groups of $1$.
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– Lee Mosher
2 hours ago
2
$begingroup$
It can be divided into equal subsets. --- I realize that you might have meant "divided" as split into half (like a sword slicing through the set only once, producing equal size parts), but in this case you're simply talking about integers divisible by $2$ and thus defining what you want to exclude in terms of itself (i.e. the argument carries little to no persuasive or rhetorical weight). I would have included this in my other comment, but ran afoul of the comment character limitations. By the way, I don't think the question is all that bad if you're just beginning to learn about primes.
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– Dave L. Renfro
2 hours ago