The Coverage Problem - thread 2

[Haydon Berrow]

Am I right in thinking that at the heart of your argument on the coverage problem is the following proto-lemma?

Proto-Lemma

Let f₁() and f₂() be functions from the ring of algebraic integers to itself and let p() = f₁() f₂() be their multiple.

Suppose that  f₁(0) = k*x for some integer k and algebraic integer x, and that f₂(0) is non-zero. If p() has sufficiently strong conditions then it follows that f₁(x) is always divisible by k in the ring. That is, for all x in the ring of abstract integers there exists algebraic integer y such that f₁(x) = k*y.

[JSH]

Ring of abstract integers? Please elaborate.

As for the rest? Used to emphasize the distributive property.

Consider the alternative with your example--factoring of k. Now then, how do you wish to split it up? Like sqrt(k)*sqrt(k)?

So my answer is: looks like you're ignoring the distributive property: k(x+y) = k*x + k*y

Which is how get very abstract things hard to relate to what I actually show. Like notice, no sums showing with your question.

There are key sums in my generalized quadratic example, and you should be relying on it too. Have posted in this group.

People arguing with me have tended to have to get to that place where they avoid the distributive property, which to me should be impossible. It's so simple!!!

___JSH

 

[Haydon Berrow]

I said we would get bogged down with me saying "I don't understand" and unfortunately I don't understand.

You wrote

Ring of abstract integers? Please elaborate.

I don't understand what you want me to elaborate upon.

As for the rest? Used to emphasize the distributive property.

I don't understand (again). Are you saying that

a) the proto-lemma is false, or

b) the proto-lemma is true but not used in your exposition, or

c) the proto-lemma is true but irrelevant, or

d) the proto-lemma is a consequence of your argument, or

e) something else?

I've snipped what you wrote after this because I wanted to get it clear in my mind before we go further.

[JSH]

On Monday, November 13, 2017 at 9:25:58 AM UTC-5, Haydon Berrow wrote:

I said we would get bogged down with me saying "I don't understand" and unfortunately I don't understand.

You wrote

Ring of abstract integers? Please elaborate.

I don't understand what you want me to elaborate upon.

To my knowledge is my first hearing of phrase--ring of abstract integers.

I talk of integer-like numbers, and there is the ring of integers, and there is the ring of algebraic integers.

Do you have a reference for THAT phrase? 

As for the rest? Used to emphasize the distributive property.

I don't understand (again). Are you saying that

a) the proto-lemma is false, or

Is not proven.

 

b) the proto-lemma is true but not used in your exposition, or

Is not proven, and is NOT used in my mathematical arguments. 

c) the proto-lemma is true but irrelevant, or

Is not proven.

 

d) the proto-lemma is a consequence of your argument, or

Has nothing to do with my math.

 

e) something else?

Is a mystery to me! But you have gone on and on about it. 

I've snipped what you wrote after this because I wanted to get it clear in my mind before we go further.

Ok. Am guessing you're focused on prior arguments I've had with people who focused on values at x = 0 versus other values.

Um, that's ancient history. You need to update to latest arguments where do not even bother with that concern, ok?

For instance, consider my generalized quadratic factorization.

Thankfully could simplify over time. Which is such a great thing! Prior arguments in this area to which you may have become attached are irrelevant.

I DO establish norms for key functions at that value, as my g's in current argument are normalized, while the f's are NOT. Otherwise? Don't care.

Which IS important.

___JSH 

[Haydon Berrow]

On Monday, November 13, 2017 at 2:38:48 PM UTC, JSH wrote:

On Monday, November 13, 2017 at 9:25:58 AM UTC-5, Haydon Berrow wrote:

I said we would get bogged down with me saying "I don't understand" and unfortunately I don't understand.

You wrote

Ring of abstract integers? Please elaborate.

I don't understand what you want me to elaborate upon.

To my knowledge is my first hearing of phrase--ring of abstract integers.

I talk of integer-like numbers, and there is the ring of integers, and there is the ring of algebraic integers.

Do you have a reference for THAT phrase? 

You are quite right. It was a senior moment by me. I meant algebraic integers.

As for the rest? Used to emphasize the distributive property.

I don't understand (again). Are you saying that

a) the proto-lemma is false, or

Is not proven.

 

b) the proto-lemma is true but not used in your exposition, or

Is not proven, and is NOT used in my mathematical arguments. 

c) the proto-lemma is true but irrelevant, or

Is not proven.

 

d) the proto-lemma is a consequence of your argument, or

Has nothing to do with my math.

 

e) something else?

Is a mystery to me! But you have gone on and on about it. 

I've snipped what you wrote after this because I wanted to get it clear in my mind before we go further.

Ok. Am guessing you're focused on prior arguments I've had with people who focused on values at x = 0 versus other values.

Um, that's ancient history. You need to update to latest arguments where do not even bother with that concern, ok?

 

For instance, consider my generalized quadratic factorization.

Thankfully could simplify over time. Which is such a great thing! Prior arguments in this area to which you may have become attached are irrelevant.

I DO establish norms for key functions at that value, as my g's in current argument are normalized, while the f's are NOT. Otherwise? Don't care.

Which IS important.

___JSH 

Ah, Ok, I'm glad that's cleared up. It's irrelevant whether it's true or not and you won't need it.

[JSH]

On Monday, November 13, 2017 at 10:03:05 AM UTC-5, Haydon Berrow wrote:

On Monday, November 13, 2017 at 2:38:48 PM UTC, JSH wrote:

On Monday, November 13, 2017 at 9:25:58 AM UTC-5, Haydon Berrow wrote:

I said we would get bogged down with me saying "I don't understand" and unfortunately I don't understand.

You wrote

Ring of abstract integers? Please elaborate.

I don't understand what you want me to elaborate upon.

To my knowledge is my first hearing of phrase--ring of abstract integers.

I talk of integer-like numbers, and there is the ring of integers, and there is the ring of algebraic integers.

Do you have a reference for THAT phrase? 

You are quite right. It was a senior moment by me. I meant algebraic integers.

Oh, ok. No problem. Probably should be a phrase actually. Maybe it is but that's digressing a bit. 

As for the rest? Used to emphasize the distributive property.

I don't understand (again). Are you saying that

a) the proto-lemma is false, or

Is not proven.

 

b) the proto-lemma is true but not used in your exposition, or

Is not proven, and is NOT used in my mathematical arguments. 

c) the proto-lemma is true but irrelevant, or

Is not proven.

 

d) the proto-lemma is a consequence of your argument, or

Has nothing to do with my math.

 

e) something else?

Is a mystery to me! But you have gone on and on about it. 

I've snipped what you wrote after this because I wanted to get it clear in my mind before we go further.

Ok. Am guessing you're focused on prior arguments I've had with people who focused on values at x = 0 versus other values.

Um, that's ancient history. You need to update to latest arguments where do not even bother with that concern, ok?

 

For instance, consider my generalized quadratic factorization.

Thankfully could simplify over time. Which is such a great thing! Prior arguments in this area to which you may have become attached are irrelevant.

I DO establish norms for key functions at that value, as my g's in current argument are normalized, while the f's are NOT. Otherwise? Don't care.

Which IS important.

___JSH 

Ah, Ok, I'm glad that's cleared up. It's irrelevant whether it's true or not and you won't need it.

Yup. And am thinking were focused on arguments made back in older debates. I moved on, thank God. The math actually has multiple paths to prove the SAME THING.

Which makes sense, eh? When is correct.

Is hard to accept though. Skepticism is very welcome. And thanks for posting here!

See how much more efficient it is? This group is a place I have for people to talk such things out, or to present their own ideas.

James Harris 

[Haydon Berrow]

We seem to be at an impasse. The opening sentence of http://somemath.blogspot.co.uk/2015/11/non-polynomial-factorization-short.html is

It can be shown that there must exist additional numbers besides algebraic integers, which are also integer-like numbers.

and the proof starts with

 ...  P(x) is a primitive quadratic with integer coefficients ....

If neither of us knows what integer-like describes then I don't see how I can understand a proof that a number satisfies the definition. Similarly, the proof starts with a primitive quadratic so presumably this is needed but, again, if neither of us knows what it means then I don't see how I understand a proof that uses its properties.

Can you see a way forward?

[JSH]

On Tuesday, November 14, 2017 at 4:57:07 AM UTC-5, Haydon Berrow wrote:

We seem to be at an impasse. The opening sentence of http://somemath.blogspot.co.uk/2015/11/non-polynomial-factorization-short.html is

It can be shown that there must exist additional numbers besides algebraic integers, which are also integer-like numbers.

and the proof starts with

 ...  P(x) is a primitive quadratic with integer coefficients ....

If neither of us knows what integer-like describes then I don't see how I can understand a proof that a number satisfies the definition. Similarly, the proof

I explained to you. I merely gave reference to cover all the bases. Integer-like numbers behave like integers in key ways, like 4+2i has 2 as a factor, and can be coprime with other numbers like 1+3i is coprime to 3. I like using gaussian integers as easy to highlight.

Integer-like numbers are NOT rational numbers that otherwise behave like integers. They also are NOT in any way like fractions.

So 1/(1+3i) is NOT an integer-like numbers.

 

starts with a primitive quadratic so presumably this is needed but, again, if neither of us knows what it means then I don't see how I understand a proof that uses its properties.

It JUST means that there is no common factor to divide across the coefficients in this context.

Like: 175x² - 15x + 2 = 0 is primitive.

5x² - 15x + 20  = 0, is not. Understand now?

 

Can you see a way forward?

Apparently I needed to simplify with you. If I give you too much detail you not only claim to not understand, you decide I don't either!

Remarkable.

___JSH