Mathematical proof..

[Surgenator]

There many types of mathematical proofs like proof by induction, proof by contradiction, etc...

My favorite mathematical proof is proof by contradiction which was used to prove that there are an infinite number of prime numbers. It goes something like this (i'm skipping the math formalism since I don't remember it)

First you need to prove that adding 1 to a number gives a completely different set of prime numbers for the original.
1) Take a number x that is a composite of a set of prime numbers A={a1,a2,..}. e.g. 36 = 2*2*3*3
2) take any prime number a in set A. e.g. a=3
3) the ratio x/a has no remainder because a is one composition of x. e.g 36/3=(2*2*3*3)/3=(2*2*3)=18
4) lets define y=x+1 which is a composite of a set of prime numbers B={b1,b2...}. e.g. 37=36+1
5) the ratio y/a=(x+1)/a=x/a+1/a which gives a remainder of 1/a because the smallest prime number is 2. e.g. 37/3 = (36+1)/3=36/3+1/3=18+1/3
6) set B does not have the prime number a in it.
7) a was a random choice, we can choose any prime number from set A get the same result
8) Therefore, there are no prime numbers that are in set A that are also in set B.

Now we do the prove there is an infinite number of prime numbers by proof by contradiction
1) Assume there is a finite number of prime numbers
2) Make the number x which is a composite of all the prime numbers
3) Let y=x+1
4) By my earlier proof, y doesn't have any of the prime numbers x has.
5) y is then a composite of prime numbers that are not part the set of all prime numbers. Contradiction.
6) Our assumptions is wrong. There is an infinite number of prime numbers.

[Alex K]

But if I said it was a special paint for example? how would I have to describe that paint?

If you need to introduce the concept of types of paint in your proof of the four color theorem, you're introducing unnecessary overhead. It should be on a more abstract level.

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Concerning your problem with the plus one.
I think you can argue that the product of all primes smaller than x, plus one, is divided by all those primes with remainder one. Is that sufficient? (long ago)

There many types of mathematical proofs like proof by induction, proof by contradiction, etc...

---

My favorite mathematical proof is proof by contradiction which was used to prove that there are an infinite number of prime numbers. It goes something like this (i'm skipping the math formalism since I don't remember it)

First you need to prove that adding 1 to a number gives a completely different set of prime numbers for the original.
1) Take a number x that is a composite of a set of prime numbers A={a1,a2,..}. e.g. 36 = 2*2*3*3
2) take any prime number a in set A. e.g. a=3
3) the ratio x/a has no remainder because a is one composition of x. e.g 36/3=(2*2*3*3)/3=(2*2*3)=18
4) lets define y=x+1 which is a composite of a set of prime numbers B={b1,b2...}. e.g. 37=36+1
5) the ratio y/a=(x+1)/a=x/a+1/a which gives a remainder of 1/a because the smallest prime number is 2. e.g. 37/3 = (36+1)/3=36/3+1/3=18+1/3
6) set B does not have the prime number a in it.
7) a was a random choice, we can choose any prime number from set A get the same result
8) Therefore, there are no prime numbers that are in set A that are also in set B.

Now we do the prove there is an infinite number of prime numbers by proof by contradiction
1) Assume there is a finite number of prime numbers
2) Make the number x which is a composite of all the prime numbers
3) Let y=x+1
4) By my earlier proof, y doesn't have any of the prime numbers x has.
5) y is then a composite of prime numbers that are not part the set of all prime numbers. Contradiction.
6) Our assumptions is wrong. There is an infinite number of prime numbers.

---

[lifesagift]

But if I said it was a special paint for example? how would I have to describe that paint?

If you need to introduce the concept of types of paint in your proof of the four color theorem, you're introducing unnecessary overhead. It should be on a more abstract level.

---

Concerning your problem with the plus one.
I think you can argue that the product of all primes smaller than x, plus one, is divided by all those primes with remainder one. Is that sufficient? (long ago)


quote='Surgenator' pid='758785' dateline='1411681980']


here many types of mathematical proofs like proof by induction, proof by contradiction, etc...

---

My favorite mathematical proof is proof by contradiction which was used to prove that there are an infinite number of prime numbers. It goes something like this (i'm skipping the math formalism since I don't remember it)

First you need to prove that adding 1 to a number gives a completely different set of prime numbers for the original.
1) Take a number x that is a composite of a set of prime numbers A={a1,a2,..}. e.g. 36 = 2*2*3*3
2) take any prime number a in set A. e.g. a=3
3) the ratio x/a has no remainder because a is one composition of x. e.g 36/3=(2*2*3*3)/3=(2*2*3)=18
4) lets define y=x+1 which is a composite of a set of prime numbers B={b1,b2...}. e.g. 37=36+1
5) the ratio y/a=(x+1)/a=x/a+1/a which gives a remainder of 1/a because the smallest prime number is 2. e.g. 37/3 = (36+1)/3=36/3+1/3=18+1/3
6) set B does not have the prime number a in it.
7) a was a random choice, we can choose any prime number from set A get the same result
8) Therefore, there are no prime numbers that are in set A that are also in set B.

Now we do the prove there is an infinite number of prime numbers by proof by contradiction
1) Assume there is a finite number of prime numbers
2) Make the number x which is a composite of all the prime numbers
3) Let y=x+1
4) By my earlier proof, y doesn't have any of the prime numbers x has.
5) y is then a composite of prime numbers that are not part the set of all prime numbers. Contradiction.
6) Our assumptions is wrong. There is an infinite number of prime numbers.

---

[Surgenator]

Concerning your problem with the plus one.
I think you can argue that the product of all primes smaller than x, plus one, is divided by all those primes with remainder one. Is that sufficient? (long ago)

It's more important to say x+1 is not divisible by ANY of the prime numbers in x's set. I believe this is sufficient.

[lifesagift]

But we're talking about colouring a map in with red, blue, green and yellow paint..... what is 'X' in this equation?

[vorlon13]

IIRC, the computer went through thousands of permutations of maps and demonstrated the proof.

At the time there was some controversy as to whether or not having a computer do all the grunt work counted, LOL. Like you could hire someone to compost their brain doing it manually.

Wiki 'four color map problem'.

[Surgenator]

But we're talking about colouring a map in with red, blue, green and yellow paint..... what is 'X' in this equation?

Completely unrelated to the color map discussion.

[Alex K]

Concerning your problem with the plus one.
I think you can argue that the product of all primes smaller than x, plus one, is divided by all those primes with remainder one. Is that sufficient? (long ago)

It's more important to say x+1 is not divisible by ANY of the prime numbers in x's set. I believe this is sufficient.

edit:

Assume that there are finitely many primes.
If p_1...p_N are all existing prime numbers and x = p_1 * ... * p_N

x /p_i = p_1 * .... * p_{i-1} * p_{i+1} * ... * p_N

and hence

(x+1) /p_i = p_1 * .... * p_{i-1} * p_{i+1} * ... * p_N with Remainder 1

so we have constructed a number which is not divisible by any of the primes, and there's your contradiction, and there can't be finitely many.

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So paint isn't my discovery, but I challenge you to tell me how to describe my new result..

I can't really do that without knowing your argument. But usually, one would introduce an abstract set {1,2,3,4,...} standing for the colors, and maybe a set of areas {A1,A2....}, a function which sets the color f(A_i)=1...4 and a relation which defines adjacency, such that A1°A2 =0 if it is not adjacent and A1°A2=1 if they are. That's just an example of what such things typically look like in principle, I'm not saying that that's how it actually works.

A often used approach towards such a type of proof is to assume that there is one scenario where 4 colours are NOT sufficient, and then bring this to a contradiction. In other words: If you can start from the assumption that there is one set of areas where there is no four color covering scheme, and you can conclude from this a statement like 1=0 using only valid logical steps, then you've shown that the assumption is false and the four color theorem is proven.

Alternatively, one might not use the areas as the fundamental object, but the boundaries and their vertices.

Maybe a proof by induction is possible by starting with an arbitrary number of areas which can be covered by four colors and then adding areas in succession and proving that one never needs to introduce a fifth color in each single step. The problem here might be that the way to color the given areas is not unique, and adding one might not be possible for an arbitrary coloring scheme of the existing areas.

[lifesagift]

Thank you all for your patience... I am a software developer, and very logically minded, so dealt with the four colour theorem slightly differently... and came up with my idea that proves you only need 4 colours to colour any map in the universe (I might be overstating things a tad there lol).. but don't know how to translate my thoughts to a definitive "proof"..

[Alex K]

Again, be our guest