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Quadratic Reciprocity: The connection that changed number theory

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Re: Quadratic Reciprocity: The connection that changed number theory

#21

Post by bmacho »

lisper wrote: post #5
fatherzine wrote: post #4

Earlier quoted context omitted.

Why do people say '2 is a solitary outlier'? Sure, by a quirk of English there is a special word 'even' denoting 'number divisible by 2'. By that logic, every prime k is a 'solitary outlier', being the only prime divisible by k. E.g. '5 is a solitary outlier, the only fiven prime'.

And while we are at it why is 1 not a prime? It conforms to the definition.

If it conforms the definition, then 1 is a prime. This is evident.

Why would they choose a definition so that 1 is not a prime? They found it the best and most useful definition that they knew at that time.

While there are many bad definitions, concepts and names in math, I don't think prime number is an example.

Also please notice that sometimes the more useful concepts have longer definitions, especially if one limits themselves in the terms they want to define it. (That is they don't use the nicest properties to define prime numbers, but they define them as:)

  A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. [0]
This is indeed longer and more complicated than if we did not exclude 1. But the concept is (at least in the current state of math) more useful than if 1 weren't excluded. Who knows, maybe in the future math they will find a new structure or theory, where the prime numbers + {1} is a really useful concept.

[0] : https://en.wikipedia.org/wiki/Prime_number

Re: Quadratic Reciprocity: The connection that changed number theory

#22

Post by mmmmmmmike »

keithalewis wrote: post #6
joewferrara wrote: post #3

Gauss died in 1855 and his work still influences modern mathematics. Someone told me while in grad school that in number theory we’re still figuring out stuff that Gauss already knew.

His motto was "Few, but ripe." I'm sure he had no conception of throwing compute power against the wall and see what sticks.

I think that’s more about what he chose to publish. He was known for discovering things and keeping them to himself.

As I understand it, he actually first came to conjecture quadratic reciprocity after doing incredible amounts of calculation by hand and noticing the pattern.

Re: Quadratic Reciprocity: The connection that changed number theory

#23

Post by ykonstant »

mmmmmmmike wrote: post #22
keithalewis wrote: post #6

Earlier quoted context omitted.

His motto was "Few, but ripe." I'm sure he had no conception of throwing compute power against the wall and see what sticks.

I think that’s more about what he chose to publish. He was known for discovering things and keeping them to himself. As I understand it, he actually first came to conjecture quadratic reciprocity after doing incredible amounts of calculation by hand and noticing the pattern.

>he actually first came to conjecture quadratic reciprocity after doing incredible amounts of calculation by hand and noticing the pattern.

Indeed, this was Gauss's method for exploring the mathematical landscape; vast calculations were his "microscope" and "experimental apparatus". He had a passion for computation, which he applied later in his life fully in his astronomy work.

Re: Quadratic Reciprocity: The connection that changed number theory

#24

Post by tgv »

contravariant wrote: post #16
lisper wrote: post #5

Earlier quoted context omitted.

And while we are at it why is 1 not a prime? It conforms to the definition.

I did once try to run through all the definitions and their generalisations, and it basically boils down to the fact that the zero ring is not a field. This gives the following stacktrace as for why you can't make 1 a prime: - A prime number is a positive prime element of Z - A prime element is an element p which is not a unit such that whenever p divides ab then p divides a or p divides b (or both) - An element whic…

The exclusion of negative numbers is not arbitrary. Depending on your exact definition, you would be left with the current prime numbers, these plus their negatives, or none at all (since 2 is divisible by -2).

Re: Quadratic Reciprocity: The connection that changed number theory

#25

Post by tromp »

tzs wrote: post #10
lisper wrote: post #5

Earlier quoted context omitted.

And while we are at it why is 1 not a prime? It conforms to the definition.

Up until sometime in the 19th century 1 was often considered to be prime. If it is considered to be a prime, then you end up with a lot of theorems that start "for all primes p != 1" or "for all odd primes != 1". If it is not considered to be a prime, you end up with other theorems that start "for all primes and 1" or "for all odd primes and 1". There are more really important theorems in the former set than the latt…

Are there any in the latter set?

Re: Quadratic Reciprocity: The connection that changed number theory

#26

Post by bonzini »

bmacho wrote: post #21
lisper wrote: post #5

Earlier quoted context omitted.

And while we are at it why is 1 not a prime? It conforms to the definition.

If it conforms the definition, then 1 is a prime. This is evident. Why would they choose a definition so that 1 is not a prime? They found it the best and most useful definition that they knew at that time. While there are many bad definitions, concepts and names in math, I don't think prime number is an example. Also please notice that sometimes the more useful concepts have longer definitions, especially if one lim…

It only makes sense to define factorization on numbers that don't have an inverse, otherwise you can make infinite factorizations of a number by multiplying by a number that has an inverse and then by its inverse.

Therefore it makes sense to classify numbers as either primes (they are their own unique factorization), or composites (they are the result of the multiplication of other numbers, and the factorization may or may not be unique), or units (numbers that have an inverse). 1 is the only unit of the natural numbers.

If you add negative integers, this results into all negated primes being also primes, and there is no unique factorization (for example 6 is 2*3 but also -2*-3).

Re: Quadratic Reciprocity: The connection that changed number theory

#27

Post by kevinventullo »

contravariant wrote: post #16
lisper wrote: post #5

Earlier quoted context omitted.

And while we are at it why is 1 not a prime? It conforms to the definition.

I did once try to run through all the definitions and their generalisations, and it basically boils down to the fact that the zero ring is not a field. This gives the following stacktrace as for why you can't make 1 a prime: - A prime number is a positive prime element of Z - A prime element is an element p which is not a unit such that whenever p divides ab then p divides a or p divides b (or both) - An element whic…

Why stop there; why can’t we consider the zero ring a field? It has addition, subtraction, multiplication, and (vacuously!) every nonzero element has a multiplicative inverse.

I realize the definition of fields usually includes the axiom 0 != 1, but I think this is just as arbitrary as “1 is not prime”.

You can even extend this to scheme theory, where Spec of the zero ring is the empty set, and conversely the zero ring is the ring of regular functions on the empty set. Note that the empty set is always open in the Zariski topology!

Anyway I’m not actually seriously advocating for this position, just that it does seem to be consistent.

Re: Quadratic Reciprocity: The connection that changed number theory

#28

Post by contravariant »

tgv wrote: post #24
contravariant wrote: post #16

Earlier quoted context omitted.

I did once try to run through all the definitions and their generalisations, and it basically boils down to the fact that the zero ring is not a field. This gives the following stacktrace as for why you can't make 1 a prime: - A prime number is a positive prime element of Z - A prime element is an element p which is not a unit such that whenever p divides ab then p divides a or p divides b (or both) - An element whic…

The exclusion of negative numbers is not arbitrary. Depending on your exact definition, you would be left with the current prime numbers, these plus their negatives, or none at all (since 2 is divisible by -2).

The notion of 'negative' doesn't transfer too well to other rings. It vaguely makes sense to only look at the natural numbers (excluding 0), primality predates negative numbers as a concept after all, but from the point of view of the more general definition it is mostly arbitrary.

Re: Quadratic Reciprocity: The connection that changed number theory

#29

Post by contravariant »

kevinventullo wrote: post #27
contravariant wrote: post #16

Earlier quoted context omitted.

I did once try to run through all the definitions and their generalisations, and it basically boils down to the fact that the zero ring is not a field. This gives the following stacktrace as for why you can't make 1 a prime: - A prime number is a positive prime element of Z - A prime element is an element p which is not a unit such that whenever p divides ab then p divides a or p divides b (or both) - An element whic…

Why stop there; why can’t we consider the zero ring a field? It has addition, subtraction, multiplication, and (vacuously!) every nonzero element has a multiplicative inverse. I realize the definition of fields usually includes the axiom 0 != 1, but I think this is just as arbitrary as “1 is not prime”. You can even extend this to scheme theory, where Spec of the zero ring is the empty set, and conversely the zero ri…

Yeah, best argument I can come up with is that the multiplication over the zero ring minus 0 does not form a group. But you'd need to modify the usual definition of a field for this to look natural.

There is quite a lot of research on a zero field, which shows up sometimes (never as a conceret object, but more like A_p is to the field of p elements what A_0 is to the zero field). Apparently you can't just use the zero ring, though the reasons why are beyond my pay-grade.

Re: Quadratic Reciprocity: The connection that changed number theory

#30

Post by gizmo686 »

tgv wrote: post #24
contravariant wrote: post #16

Earlier quoted context omitted.

I did once try to run through all the definitions and their generalisations, and it basically boils down to the fact that the zero ring is not a field. This gives the following stacktrace as for why you can't make 1 a prime: - A prime number is a positive prime element of Z - A prime element is an element p which is not a unit such that whenever p divides ab then p divides a or p divides b (or both) - An element whic…

The exclusion of negative numbers is not arbitrary. Depending on your exact definition, you would be left with the current prime numbers, these plus their negatives, or none at all (since 2 is divisible by -2).

Primality is defined for any ring (although some rings have no primes). If you are asking about prime integers, then negative primes are themselves prime. Most of what you know of primality still holds if you add 'up to unit multiples' to all of your theroums.
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