Numbers and Logic

[arfa brane]

I'll try to avoid using the word "science" in this thread (which of course, doesn't mean someone else can use it).

Anyways, about numbers and logic: why does a set of symbols, the Arabic numerals, have a logic that we can't write an explicit formula for--prime numbers--and why is primality defined by division or a set of divisors?

So we can write: If n is prime, then the divisors of n are 1 and n, and no other numbers.
Since it's symmetric we also have: If n and 1 are the only divisors of n then n is prime.

This is more or less useless if we have a number, n, and want to know if it's prime. We need more logic!

 

[arfa brane]

Should add that, in number theory, a number is an integer.

Is number theory a science of numbers? Is it a logic of numbers? Does it matter, since "number theory" is accepted language?
If there is no science of logic, I think the theory here based on prime numbers is in trouble. Bigly.

 

[TheFrogger]

What about negatives of a prime?

 

[river]

Why do the numbers and logic matter ?

 

[TheFrogger]

Logic matters because it supplies rules and structure to a creation. Everything thereafter must follow these rules, else be negated.

 

[river]

Logic matters because it supplies rules and structure to a creation. Everything thereafter must follow these rules, else be negated.

Does reason matter ?

 

[TheFrogger]

Reason is why something happens. The reason.

Reason matters because it is common sense. We all understand it. It is a common language, like mathematics.

 

[arfa brane]

Number theory says every number (every integer) is prime or a product of primes. This leads immediately to the problem of the numbers 0, and 1.
So if you just tighten the theory so it applies to numbers with absolute value 2 or more; you exclude the problem numbers for now and focus on the numbers where it holds.

With the fact that a prime is divisible by itself, and trivially by 1 (but so is anything), we can define a division algorithm (a thing with steps in it). In the case of numbers we get algorithms, or algorithmic structure (counting is one of these for instance). Call this structure a logic, or an algebra if you like.

 

[DaveC426913]

I'm not really sure if you're asking a question.

 

[arfa brane]

I'm not really sure if you're asking a question.

. . . why does a set of symbols, the Arabic numerals, have a logic that we can't write an explicit formula for--prime numbers--and why is primality defined by division or a set of divisors?

That's actually two questions.

 

[DaveC426913]

. . . why does a set of symbols, the Arabic numerals, have a logic that we can't write an explicit formula for--prime numbers

Pretty sure the Arabic symbols or numbers don't have anything to do with why we have no formula. Pretty sure it's the same thing no matter what counting system - or even base - we use.

Question: what would you expect from this formula? The series of prime numbers is infinite. Not sure a formula can produce an infinite set.
Or are you looking to input a specific number and have the formula output prime/not?

Maybe an algorithm is what you're looking for.

--and why is primality defined by division or a set of divisors?

Why wouldn't it be?
The definition of primality explicitly invokes multiplication - and division is defined as the inverse of multiplication.
So, if it weren't defined by division, it would be ... something other than primality.

 

[arfa brane]

Pretty sure the Arabic symbols or numbers don't have anything to do with why we have no formula. Pretty sure it's the same thing no matter what counting system - or even base - we use.

I'm pretty sure numbers do have something to do with this formula. The Arabic numerals are symbols, we interpret them as numbers but as you say, it doesn't matter what the number system is.

But definitely, the OP assumes there is at least one number system (the Arabic numerals and base 10), The question only makes sense when numbers are defined.

Question: what would you expect from this formula?

That I can't find it, or any other formula that says any number is prime or not.

Which isn't to say I can't write a program that will tell me; what I mean by a formula is a closed-form expression I can plug any number into and have a result.
The formula is not an algorithm, however. Moreover, nobody has found such a formula although there are several algorithmic methods to determine if a number is prime.

 

[DaveC426913]

I'm pretty sure numbers do have something to do with this formula.

I meant "Arabic symbols or Arabic numbers".

Which isn't to say I can't write a program that will tell me; what I mean by a formula is a closed-form expression I can plug any number into and have a result.
The formula is not an algorithm, however. Moreover, nobody has found such a formula although there are several algorithmic methods to determine if a number is prime.

Ever more efficient ways of finding prime numbers this is a very active area of research. Were you to come across such a formula, you'd likely win a Nobel prize.

 

[James R]

Anyways, about numbers and logic: why does a set of symbols, the Arabic numerals, have a logic that we can't write an explicit formula for--prime numbers--and why is primality defined by division or a set of divisors?

It's not the only way we could define those numbers, although it is useful.

If you want an explicit procedure for generating prime numbers, there's Aristothenes' sieve, for starters.

 

[arfa brane]

It's not the only way we could define those numbers, although it is useful.

If you want an explicit procedure for generating prime numbers, there's Aristothenes' sieve, for starters.

But aren't you just saying the prime numbers can be defined algorithmically?

The takeaway being that there are at least two kinds of definition: one based on a set of divisors for any number, the other being a procedural definition.
The first kind doesn't lead to a closed-form solution for n (any number).

That is, if you see a difference between algorithmic definitions, and functional definitions.

 

[James R]

The first kind doesn't lead to a closed-form solution for n (any number).

I'm not sure what you mean by that.

Any number, n, has a particular set of divisors. Isn't it a closed-form solution to determine the divisors to see if it has more than two?

 

[arfa brane]

I'm not sure what you mean by that.

Any number, n, has a particular set of divisors. Isn't it a closed-form solution to determine the divisors to see if it has more than two?

Think about writing a formula that determines the set of divisors of any n. So this formula will tell you n is prime when the number of divisors is two.

But that's equivalent to a formula that tells y0u the kth prime, for any k.
So there should be a formula that determines the first k primes. So let's see what it looks like for say, the first 10 primes, should be a doddle.

 

[James R]

So there should be a formula that determines the first k primes. So let's see what it looks like for say, the first 10 primes, should be a doddle.

I'm not a mathematician, but as far as I'm aware there is no such formula. If there was, then we wouldn't need computational projects like the Great Mersenne Prime Search.

There are lots of unsolved problems with primes. I have an entire book that talks about their relationship to the Riemann zeta function and the Riemann hypothesis (which still isn't proven), about half of which is beyond my level of mathematical ability to understand. Then there are the famous conjectures, like the Goldbach conjecture and the twin prime conjecture, which have seen some progress towards proof but no proof yet.

Prime numbers are one of those areas of maths that I find fascinating. There seems so much structure to prime numbers - hidden relationships, as well as important results that rely on certain facts (or conjectures) about primes. They are a prime (excuse the pun) example of the unpredictable patternicity of mathematics. If I was religious, I'd be inclined to put it down to God.

 

[iceaura]

If there is no science of logic, I think the theory here based on prime numbers is in trouble. Bigly.

Science would introduce uncertainty - it's empirical, and does not deal in proof.
Whatever is meant by a "theory based on prime numbers", it is necessarily a mathematical theory and deals in proof; and that is what keeps it out of trouble. Proof is the foundation of mathematics, and it's not available to science.

There are many different proofs of the Prime Number Theorem - http://mathworld.wolfram.com/PrimeNumberTheorem.html
https://en.wikipedia.org/wiki/Prime_number_theorem (note that these two web sources do not duplicate each others list of named proofs - and together they are not complete).
as far as the thread topic, note the significant role played by very large numbers in dealing with primes and their properties: the unusual number of failed hypotheses and false conjectures by very good mathematicians regarding primes seems to be a consequence of extrapolating from what is observed in small numbers ( where by "small" is meant something like "can be written as an integer on one page of paper").

That historical experience has taught wariness.

 

[el es]

There is no logic to Arabic numbers( symbols). 2 represents a set of 2 ones, 3 represents a set of 3 ones, 4 represents a set of 4 ones etc.

Find a prime using a primitive system using only one digit and sets of one digit. Complete sets means it is not a prime.

[11]+[11]=1111 not a prime. 2+2=4

[11]+[11]+1=11111 is a prime, 1 can't be used as a set. 2+2+1=5

[1111]+[1111]=11111111 4+4=8

[111]+[111]+[111]=111111111 not a prime 3+3+3=9


[11111]+[11111]=1111111111 is not a prime 5+5=10

[11111]+[11111]+1=11111111111 is a prime 5+5+1=11