### Monday, January 05, 2009

## JSH: Simple proof, but social response

I have found a remarkable error in established number theory which has physics implications, but while it's VERY easy to prove the error, mathematicians are invested fully in it, and have resisted efforts to get them to acknowledge mathematical proof.

The error is easily shown with a simple construction on the complex plane:

7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2)

where b_1(0) = 0, b_2(0) = 0, a_1(x) = b_1(x), and

a_2(x) = b_2(x) - 1, and the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0.

Now you have NON-LINEAR functions.

Explicitly you have

a_1(x) = ((7x - 1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x - 1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

The result on the complex plane shows that 7 distributed through only one of the a's, but which one?

That's not visible because the square roots are not in general resolvable. Consider:

x^2 + 4x + 3 = 0, and with x=(-4 +/- sqrt(16 - 12))/2 = (-4 +/- sqrt(4))/2, which root has 3 as a factor? You can't say until you take the square root.

Despite all of the above, the reality of the proof does not seem to crystallize for most people without another example so consider:

7*(x^2 + 3x + 2) = (7x + 7)(x + 2)

I can let that be

7*(x^2 + 3x + 2) = (f_1(x) + 7)(f_2(x) + 2)

where f_1(x) = 7x, and f_2(x) = x.

And those are LINEAR functions.

There is not a mathematician in the world who will argue with you about whether or not 7 distributed through the factor

(f_1(x) + 7)

given

7*(x^2 + 3x + 2) = (f_1(x) + 7)(f_2(x) + 2), where f_1(x) = 7x, and f_2(x) = x,

with linear functions.

Notice though that with 7(x+1) = 7x + 7, you simply have the distributive property:

a*(b + c) = a*b + a*c

With b a linear function—no argument.

e.g. a*(f(x) + b) = a*f(x) + a*b

when f(x) is a linear function with

7*(x^2 + 3x + 2) = (f_1(x) + 7)(f_2(x) + 2)

no argument from the mathematical community.

But with f(x) a NON-LINEAR function i.e.

7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2)

where b_1(0) = 0, b_2(0) = 0, a_1(x) = b_1(x), and

a_2(x) = b_2(x) - 1, and the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0, so

a_1(x) = ((7x - 1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x - 1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

now there is an uproar from the mathematical community as the result now follows that only one of the roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0

is a product of 7 and some other number, which leads to a conclusion in other rings.

How?

Well consider again the simple example: f(x) = 7x

That is TRUE on the complex plane, but the equation is ALSO valid in the ring of integers, where it gives a factorization result.

a^2 - (7x-1)a + (49x^2 - 14x) = 0, with solutions

a_1(x) = ((7x - 1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x - 1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

is true in the ring of algebraic integers, but it can be proven that NEITHER root can have 7 as a factor in that ring if the a's are not integers for an integer x.

(If the a's are integers, like with x=0, then exactly one root has 7 as a factor in that ring as expected from the result on the complex plane.)

The key point of all the discussion above is that the distributive property is key here, where with

a*(f(x) + b ) = a*f(x) + a*b

the type of function shouldn't matter.

With a linear function f(x)—no debate.

With a non-linear function, furious debate.

But how does this relate to physics?

Well physicists were never "pure" in that their results are usually expected to work in the real world (ignore "string theory" for the moment), but mathematicians around the turn of the 20th century strongly endorsed "pure mathematics" known for not giving practical results, but they promised that in time the mathematics might be of value in practical areas.

But what if it was wrong?

This result indicates that over a hundred years of results in modern number theory are in fact, wrong, because it shows a contradiction between a key element in such theories—the ring of algebraic integers—and the field of complex numbers.

Quite simply, the field of complex numbers with the distribution argument, disagrees with the ring of algebraic integers, where 7 cannot be a factor, when the a's are non-rational with integer x.

There is a fight between the dominant field for physicists—the complex plane—and the dominant ring for number theorists—the ring of algebraic integers.

It is an unfortunate duel to the death for theories dependent on one position or the other.

Either the complex plane is correct and theories based on its correctness are correct, or the mathematicians are correct, and their research using algebraic integers wins.

BUT physicists work in the real world. Number theories more often work in the "pure math" arena.

So there is no real debate. They are wrong, the physicists are right. The complex plane wins and number theorists have a hundred years of error.

But being proud they resist the truth like so many people before them.

Wishing they were right, they betray their own field, hide from the truth, and continue to teach false ideas to young minds, despite one of the easiest proofs of error possible—for a problem that has lurked for over one hundred years.

Truth IS stranger than fiction. As mathematicians betray their field to hold on to error, history records one of the greatest intellectual challenges in the history of the human species.

It is a battle for the very heart of mathematics itself.

The error is easily shown with a simple construction on the complex plane:

7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2)

where b_1(0) = 0, b_2(0) = 0, a_1(x) = b_1(x), and

a_2(x) = b_2(x) - 1, and the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0.

Now you have NON-LINEAR functions.

Explicitly you have

a_1(x) = ((7x - 1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x - 1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

The result on the complex plane shows that 7 distributed through only one of the a's, but which one?

That's not visible because the square roots are not in general resolvable. Consider:

x^2 + 4x + 3 = 0, and with x=(-4 +/- sqrt(16 - 12))/2 = (-4 +/- sqrt(4))/2, which root has 3 as a factor? You can't say until you take the square root.

Despite all of the above, the reality of the proof does not seem to crystallize for most people without another example so consider:

7*(x^2 + 3x + 2) = (7x + 7)(x + 2)

I can let that be

7*(x^2 + 3x + 2) = (f_1(x) + 7)(f_2(x) + 2)

where f_1(x) = 7x, and f_2(x) = x.

And those are LINEAR functions.

There is not a mathematician in the world who will argue with you about whether or not 7 distributed through the factor

(f_1(x) + 7)

given

7*(x^2 + 3x + 2) = (f_1(x) + 7)(f_2(x) + 2), where f_1(x) = 7x, and f_2(x) = x,

with linear functions.

Notice though that with 7(x+1) = 7x + 7, you simply have the distributive property:

a*(b + c) = a*b + a*c

With b a linear function—no argument.

e.g. a*(f(x) + b) = a*f(x) + a*b

when f(x) is a linear function with

7*(x^2 + 3x + 2) = (f_1(x) + 7)(f_2(x) + 2)

no argument from the mathematical community.

But with f(x) a NON-LINEAR function i.e.

7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2)

where b_1(0) = 0, b_2(0) = 0, a_1(x) = b_1(x), and

a_2(x) = b_2(x) - 1, and the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0, so

a_1(x) = ((7x - 1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x - 1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

now there is an uproar from the mathematical community as the result now follows that only one of the roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0

is a product of 7 and some other number, which leads to a conclusion in other rings.

How?

Well consider again the simple example: f(x) = 7x

That is TRUE on the complex plane, but the equation is ALSO valid in the ring of integers, where it gives a factorization result.

a^2 - (7x-1)a + (49x^2 - 14x) = 0, with solutions

a_1(x) = ((7x - 1) +/- sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

a_2(x) = ((7x - 1) -/+ sqrt((7x-1)^2 - 4(49x^2 - 14x)))/2

is true in the ring of algebraic integers, but it can be proven that NEITHER root can have 7 as a factor in that ring if the a's are not integers for an integer x.

(If the a's are integers, like with x=0, then exactly one root has 7 as a factor in that ring as expected from the result on the complex plane.)

The key point of all the discussion above is that the distributive property is key here, where with

a*(f(x) + b ) = a*f(x) + a*b

the type of function shouldn't matter.

With a linear function f(x)—no debate.

With a non-linear function, furious debate.

But how does this relate to physics?

Well physicists were never "pure" in that their results are usually expected to work in the real world (ignore "string theory" for the moment), but mathematicians around the turn of the 20th century strongly endorsed "pure mathematics" known for not giving practical results, but they promised that in time the mathematics might be of value in practical areas.

But what if it was wrong?

This result indicates that over a hundred years of results in modern number theory are in fact, wrong, because it shows a contradiction between a key element in such theories—the ring of algebraic integers—and the field of complex numbers.

Quite simply, the field of complex numbers with the distribution argument, disagrees with the ring of algebraic integers, where 7 cannot be a factor, when the a's are non-rational with integer x.

There is a fight between the dominant field for physicists—the complex plane—and the dominant ring for number theorists—the ring of algebraic integers.

It is an unfortunate duel to the death for theories dependent on one position or the other.

Either the complex plane is correct and theories based on its correctness are correct, or the mathematicians are correct, and their research using algebraic integers wins.

BUT physicists work in the real world. Number theories more often work in the "pure math" arena.

So there is no real debate. They are wrong, the physicists are right. The complex plane wins and number theorists have a hundred years of error.

But being proud they resist the truth like so many people before them.

Wishing they were right, they betray their own field, hide from the truth, and continue to teach false ideas to young minds, despite one of the easiest proofs of error possible—for a problem that has lurked for over one hundred years.

Truth IS stranger than fiction. As mathematicians betray their field to hold on to error, history records one of the greatest intellectual challenges in the history of the human species.

It is a battle for the very heart of mathematics itself.