Gangerolf
Last edited 1/6/2010
Thursday, October 15, 2009

Fermat's Last Theorem

Fermat's last theorem has fascinated pedestrians and professionals alike over several centuries. The theorem itself is little more than a curiosity, its attraction and fame are based on the fact that until recently it resisted all attempts at finding a rigorous proof, and in the process spawned new approaches to number theory. Finally, Wiles and Taylor provided a proof  in 1994. It is long and complicated, if Fermat had a proof, this was not it.

If!  - Most, if not all, writers express doubts that he had a proof. The argument goes that the famous annotation in his book was written around 1630 and he was striving mightily thereafter to find proof for special cases. If he had a general proof, why bother with special cases? It doesn't make sense.

It doesn't, unless the time line is wrong. We do not know the date of the annotation; it could have been written much later. Suppose he struggled with the issue in the 1630s and gave up, to devote his time to more rewarding quests. In 1653 Pascal went public with his observations on the marvelous  triangle that bears his name. In that period Fermat corresponded with Pascal on the theory of chance. He may have seen  that Pascal's triangle offered the key to solve his old problem. He wrote the jubilant annotation, and, problem solved, he lost interest. The foundation of probability calculus was far more interesting.

**********

The Equation

aⁿ + bⁿ =c ⁿ

has no non-zero integer solution when n > 2.If there is no integer solution when when a, b, and c are mutual primes and n is an odd prime, no other integer solution is possible. The special case, n is a power of 2, was proven by Fermat.

*********

Given Pascal's Triangle;

the n'th line provides the coefficients for expanding the expression

(b+x)ⁿ

When n is prime, n divides all coefficients greater than 1.
Set c = b+x; x being integer, x < a
then:

aⁿ  + bⁿ= (b+x)n = bⁿ + nb^n-1x + … … + nbx^n-1 + xⁿ

and:

aⁿ = nb^n-1x + … … + nbx^n-1 +xⁿ

**********

Show that a and n are coprime:

If n divides a, it must also divide x and it cannot divide b,

aⁿ =  nb^n-1x + (n(n-1)/2)b^n-2x² … + nbx^n-1 + xⁿ

If n divides a, divide both sides by n³ to get

aⁿ /n³ =  nxb^n-1/n³ + {(n(n-1)/2)b^n-2x² … + nb x^n-1 +  xⁿ}/n³

Then if n divides a, an integer left side is equated to a non-reducible rational right, n and a must be  coprime; Fermat's Little Theorem is applicable.

Show that x does not divide a:

(x does also not divide b. If it did, b and c would not be coprime.)

aⁿ  =  nb^n-1x + … … + nbxn-1 + xⁿ

then aⁿ /x is integer.
Although x divides aⁿ ; unless x is prime it does not necessarily divide a.
If x divides a, divide by x². Then

aⁿ /x² =  nb^n-1 /x + … … + nbx^n-3+x^n-2

If x divides a, an integer on the left side is equated with  a fraction on the right.

Based on our premises, then

aⁿ  ≡  (xⁿ)Mod (n)

We know, from Fermat's "Little Theorem" that

aⁿ  ≡ a Mod (n)

We have  x < a. Divide by a:

a^n-1  ≡ 1 Mod (n)

and

  a^n-1  ≡ (xⁿ)/a Mod (n)

leading to a contradiction: If a = xⁿ, then x must divide a, contrary to the demonstration above,

 x = 1 is not a valid result.

Fermat's "Little Theorem"

If p is a prime number and a a natural number, then

   a^P ≡ a (mod p)                                                        (1)

Furthermore, if p ≠ a (p does not divide a ), then there exists some smallest exponent d such that

a^d ≡ 0(mod p)                                                   (2)

and d  divides p-1 . Hence,

a^p-1 ≡ 0(mod p)                                                   (3)

                                                                                                   (Wolfram)                                                                                                                                                                      

 Pierre de Fermat first stated the theorem in a letter dated October 18, 1640 to his friend and confidant Frénicle de Bessy as the following[1] : p divides a^p-1 − 1 whenever p is prime and a is coprime to p                                                                                                      (Wikipedia)

The converse is not always true. Some exponents, although they are composite numbers,  may satisfy the congruence. They are called pseudoprimes.

The special case: a = 2 is nicely illustrated in Pascal's triangle, The sum of the coefficients in the n'th line is 2ⁿ. If n is prime we have

2n ≡ 2 (mod n)

The smallest pseudoprime to base 2 is 341. It does not divide all the  coefficients  >1 on its line in Pascal's triangle.

Comment:
Ever since I got bit by the bug I have had several "mirabilem" proofs that all have fizzled, No doubt the attempt above will suffer the same fate. I shall be in good company; it was fun (and educational) trying,

These were tools at Fermat's disposal. He did not have modular arithmetic but the arguments can be made without its help.

The case n = 2  is special because the polynomials are reduced to simply 2bx + x².

a²  = 2bx + x².

It has nontrivial integer solutions if x  = 1. a  is then odd; any odd value for a will yield a solution,

Pedestrian Physics 14

Redshift

A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.
(Max Planck)

The vorpal universe is a many-splendored thing with pirouetting galaxies dancing to the music of the spheres. with maelstroms sucking in energy and matter, destroying and reshaping, dragging the products into the vorpal interior whence they resurface, buoyed by the inflationary forces; but unless the vorpal itself spins, there is no systemic motion. The cosmic redshift phenomenon must therefore be a function of position, not motion.

With few data as a guide, it is a field wide open for speculations. I shall assume that photons follow the curve of the great circle combining source and target. It means, as the voyage progresses, the momentum vector must rotate to be aligned with the tangent.

Rotation requires work and is best described by introducing the concept of torque, ‘τ’:

E = τϴ
E is work done
τ is torque
ϴ is angle rotated

There is no outside energy available, all forces are in balance. The work therefore must be done at the expense of the photon itself.

What causes the energy loss? There is no known equivalent of friction to slow down a photon, A different explanation is available: the extinction theorem. As it applies to photons in space, when a photon interacts with a particle, it is extinguished. If the particle cannot hold on to it, it must eject a new photon. This photon must be ejected with speed ‘c’ relative to the particle. In the process total momentum must be conserved and, depending on the situation, energy may be lost to the intercepting particle. The new photon will have speed ‘c’ but a reduced momentum. Its average speed will remain close to ‘c’ but after death by a thousand cuts, a tired replica of the original photon will arrive. Hence the cosmic redshift.

This scenario permits us to shift the focus to the central angle. Since the linear velocity remains near 'c', the angular velocity is constant, and equal to the rate of rotation of the momentum.

Stipulate that at each encounter the loss is proportional to the arriving mass, and smooth the curve:
dm = -mkdϴ
d(ln(m₀/m_ϴ ))= kdϴ
z + 1 = m₀/m_ϴ = e^kϴ

Set z = 1000 when ϴ = π, then k = 2.2.

The Virgo cluster, at a distance of 50 Mly, is at the limit of distance estimates based on luminosity. The Hubble z-value is listed as 0.004. It corresponds to a central angle in the vorpal model: ϴ = 0.002 radians. With these numbers, the estimate for R, the vorpal radius, is:

R = 25.000 Mly

Luminosity
Plot: y = e^2.2ϴ - 1

Redshift is all about the quality of the arriving photons, luminosity is also about the number of photons arriving.

The inverse square law makes sense in a “flat” three-dimensional space. In the curved skin of the vorpal it works only in close proximity to the source. The density of photons will be reduced more slowly than in flat surroundings and reach its minimum when the central angle is π/2. The photons will then start to converge and arrive exhausted but in full force at the antipode.





The 'cosmic background radiation' is reported to have z-values in excess of z = 1000. In the vorpal model it implies that the origin is close to our antipode. The anisotropy makes sense if the photons are emitted from individual galaxies.

The lower plot shows that out to 200 Mly, or so, it will probably be impossible ever to obtain data that can differentiate the exponential curve from Hubble's linear correlation.

Bread on the water 6

The First Law

It is often said that Newton’s first law of motion is redundant. It is just a special case of the second law.This ignores the historical context. Until Newton put his quill to the parchment, “Natural Motion” was a circle. Even the Great Galileo went to his grave, believing this to be true. He espoused the cycles and epicycles of the Copernican cosmos but Keppler’s ellipses were anathema.

The straight line of the law was the last nail in the coffin of Aristotelian physics. The logical extension of calling it redundant would be to declare all three laws redundant. They are “just” special cases of the law of conservation of momentum.

Bread on the water 4

In all this they are not seeking for theories and causes to account for observed facts, but rather forcing their observations and trying to accommodate them to certain theories and opinions of their own. (Aristotle)

The Exploding Universe

“The majority is never right!”

So says Dr. Stockman in Henrik Ibsen’s “An Enemy of the People”. It is also a scientific truth. You can read the history of science on the tombstones in its graveyard.

The majority now agrees that the Universe is expanding at an increasing rate.

So it cannot be right!

My main objections to the theory are that the foundation is weak, and the development of the theory is unconvincing:

1. Light is moving through the university at a constant speed, regardless of its source.
2. The "Relativistic Doppler Effect" changes the quality of the photon.
3. The redshift is due to the relativistic Doppler effect.
4. Hubble's Law is valid at any distance
5. "Time", a dimensionless concept, is transmogrified to a space dimension.
6. The time-line of the space development is extrapolated back to "The Beginning of Time"
   

1. Einstein's second postulate of relativity is still short of experimental verification. The "expanding Universe" needs it but does not justify it. The Ritz postulate together with the extinction theorem will also predict a constant velocity at or close to c (ignoring the time lost to photon/particle interactions).

2. The relativistic Doppler effect is logically implausible. The space of the universe has no medium capable of transmitting an impulse. The mass value of a photon should not depend on an arbitrary choice of reference frame.

3. That the redshift is due to this Doppler effect is an inference, not a proven fact.

4. Hubble's Law is empirical. Within a radius of 65 light years we have observational data on distance, based on parallax measurements, and, independently, records of the redshift. About 2000 stars are estimated to be within this range. The total number of stars in the universe is said to be 8*10²². The expanding universe model assumes that Hubble’s law, derived from this small non-random sample (and confirmed by other means out to 60 Mly), applies to the entire Universe.

6. The extrapolation back to "The Beginning of Time" is an operation fraught with peril:

In Norway there is a universal duty to military service. The Norwegian Statistical Central Bureau has data on recruits over the last 150 years or so. The average height has increased steadily from decade to decade.


Kjell Aukrust, (author and cartoonist), extrapolated these data back to the year 1000. He concluded that the Vikings were about two inches tall. “If you had been present a Stiklestad”, he said, “you would not have seen the battle. You might have heard the rustle in the grass.”

In the case of the expanding universe it is not funny. The time line ends in a singularity, in nothing. From this nothing, scientists believe that the universe was created with a bang. It has kept expanding ever since.

Yet, nothing comes from nothing. There are depths in nature that we cannot fathom. We still don't know what motion is but at least we know the limitation of our knowledge. We are a long way from saying the same about the universe.

Bread on the water 3

The vorpal (redux)

If the universe is the skin of a vorpal, how thick is it?

The first answer is: Extremely thin! The fourth dimension has a magnitude too small to be registered by our finest instruments so far.

A clue may be available in the in the so-called tunnelling phenomenon. Subatomic particles in radioactive substances are able to escape impenetrable energy barriers .

The largest such particle is the alpha particle. If, instead of drilling through the barrier, the particles “jump the fence”, their size might be an indication of the skin thickness.

The vorpal is timeless. It has no questions, no answers, about beginning and end. Individual stars mau be born and stay until they are weary and sick of shining but the vorpal itself, like Ol' Man River, just keeps rolling along.

Science is about the here and now, the past and the future but not about the origin, not about the end. Those are questions for mythology and religion. Science deals with  things we can know, faith is a firm conviction about things we cannot know.

Zero, by definition, does not exist, and Infinity has no boundary, yet both concepts are necessary adjuncts to the vocabulary of science. They are the meat and  the bread of religion.

The search for faith, the quest for knowledge, are both in our genes. When the twain meet, trouble starts.
 
There is a sweet irony in the vorpal model, if true. What we consider a straight line is in the model a segment of a vorpal's great circle.  Then the old sages have the last laugh; natural motion is a circle.

Bread on the water 2

Arrogance

Science is pursued by an elite
Elitism breeds arrogance, it is a trap scientists must avoid
They ignore it at their peril
The antidote is a dose of Hippocratic humility:

Life is short, and Art long; the crisis fleeting; experience perilous, and decision difficult.

The most egregious example I have seen of disdain for hoi polloi is a statement by Parmenides:

[…] mortals knowing nothing wander aimlessly, since helplessness directs the roaming thought in their bosoms, and they are borne on, deaf and like-wise blind, amazed…

I have not yet come across a similar disdain for the common man in Aristotle’s writings. His arrogance, if you call it arrogance, is his conviction that human reasoning will not lead you astray. In this he was wrong and eventually his errors in physics and cosmology had to be corrected.

Similar sentiments are echoed even today:

"The general public may be able to follow the details of scientific research to only a modest degree, but it can register at least one great and important gain: confidence that human thought is dependable and natural law is universal." --Albert Einstein

Human reasoning is contaminated by bias and emotion. Even at the best,- and Aristotle was the best,- reasoning is at the mercy of the premises. “The most erroneous premises”, said Peer Gynt, “often lead to the most original results”.

Aristotle’s errors put a brake on the development of physics and astronomy for two millennia. It is the price you pay for blind faith in your swami.

Bread on the water 1

Random Thoughts

For after all what is man in nature? A nothing in relation to infinity, all in relation to nothing, a central point between nothing and all and infinitely far from understanding either. (Pascal)

This set of essays will be more personal and argumentative. Bear with me, - or skip.

I am, I hope, not eristic. The purpose of a discussion is not to win an argument but to seek a valid solution to a problem.

So far, with the possible exception of the deSitter essay, (Pedestrian Physics 8), I have not found arguments that settle the Ritz/Einstein controversy. Even the vorpal model can co-exist with Einstein's "Second Postulate". I prefer Ritz, because of its sanity and simplicity. If Occam's Razor is a scientific tool, then Ritz gets a clean shave while Einstein will leave the barber shop cut up and bloodied.

Comparing the vorpal model to the exploding universe leaves no room for agnosticism. It will be one or the other (or neither). The vorpal model has many loose ends but until another contender emerges it deserves to be judged on its merits. Sometimes loose ends can be tied up.

A discussion must focus on the underlying premises, inferences, assumptions and extrapolations. For the vorpal model, they were laid out in the series of Pedestrian Physics. I shall let others hammer them. I shall deal with my misgivings about the exploding model.