How significant is the fact 1 isn't a prime number? What will happen if it is? What areas of Mathematics are affected by changing the fact? I know why and how 1 isn't a prime. My question is how significant is the fact.
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2Since $1^n = 1$, there is no longer a unique prime factorization of any integer, and pretty much the whole of number theory will fall apart unless you change "prime" to "prime $\ne 1$" virtually everywhere. – alephzero Feb 18 '19 at 16:53
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6It will make theorems more annoying to state. – Randall Feb 18 '19 at 16:54
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See the answers in Why is $1$ not a prime number? – Bill Dubuque Feb 18 '19 at 16:55
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I know why and how 1 isn't a prime. My question is how significant is the fact. – Krishna Deshmukh Feb 18 '19 at 16:56
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What do you mean by "changing the fact"? Facts in math are not a matter of personal opinion. If you change the definition of a prime number to make 1 a prime, you can't then say "well, I know it isn't a prime really". – alephzero Feb 18 '19 at 17:02
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HaHa...Maybe I could've, if the definition was the changed one. – Krishna Deshmukh Feb 18 '19 at 17:08
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1It's analogous to asking what will change if we allow $0$ to be positive (as the French do). Many theorems using "positive" would need to be updated to remain correct. – Bill Dubuque Feb 18 '19 at 17:10
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It would changed the fact, that in geometry, length is the most extended dimension of an object. – LAAE Feb 18 '19 at 19:09
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@usiro: I did not get your point. – Krishna Deshmukh Feb 18 '19 at 19:29
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Basically $1$ is a unit and cannot be represented as a line. – LAAE Feb 18 '19 at 19:55
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1@usiro I can't make any sense of your remarks. In any case notions such as irreducible, prime and unit are independent of geometry. – Bill Dubuque Feb 18 '19 at 21:27
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@BillDubuque That's ok - the above are my thoughts, not the answer. Every prime greater than $7$ can be formed as a sum of two rectangles minus their intersection - therefore a prime itself cannot be formed as a rectangle. – LAAE Feb 18 '19 at 22:50
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That wouldn't be a disaster, but it would add some very annoying things. For example, instead of the uniqueness in the fundamental theorem of arithmetic we would have "uniqueness up to finitely many multiplications of $1$".
Mark
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Actually yes, we can. But even if we want to say that every integer can be represented as a finite product of primes it would still not be a unique decomposition. – Mark Feb 18 '19 at 17:04
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@Krishna No, infinite products don't make sense in general (one needs additional ring structure to make sense of them). – Bill Dubuque Feb 18 '19 at 17:16
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We need factorisation of positive integers to be unique in many contexts, and this fails if we take $1$ to be prime, since then $6,$ for example, would factorise in the infinity of ways $$2×3=2×3×\underbrace{1×1×\cdots×1}_{n},$$ where $n$ is a nonnegative integer. This happens for the other positive integers too.
Actually, $1$ is different from the other positive integers in that it is neither composite nor prime. This distinguished position follows from the fact that $1$ is the multiplicative identity in $\mathbf Z;$ that is, for any integer $m,$ we have that $$1×m=m×1=m.$$
Allawonder
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