Fermat's last theorem

[Good Dreams]

I don't remember who proved that Fermat's last theorem was right, but Fermat is said to have found the solution quite easily. Do you believe there is a simpler proof of his theorem or you think he was just bluffing?

For those who may not know:

x^n + y^n = z^n doesn't have any solution for n > 2

That's the theorem.

[Guv]

I think the prover's name is Wiles or Whiley or something like that. Using Fermat in a search is likely to come up with a site giving his name and the proof.

I believe that Fermat had a proof. The history of his life strongly implies that he at least thought he had a proof. If he was in error here, it is the only mistake he ever made in his published statements. He made other statements which defied mathematicians for a long time.

His Last Theorem is given that name because it is was the last unresolved theorem or conjecture which he made. It was not the last chronologically. When Fermat was alive, mathematicians played competitive games with each other. In order to gain an edge in these games, they kept some of their work secret.

About 100 years ago, there was a proof published and accepted as valid for a year or so. Then an incredibly subtle flaw was discovered. There is reason to believe that this proof or a similar one with a subtle flaw was the proof that Fermat considered valid.

There is evidence which suggests (but does not prove) that Fermat developed theorems and techniques not yet rediscovered by modern mathematicians. In the field of Number Theory, he was a genius. This is an obscure field which has been largely ignored by many of the great mathematicians of the last 100 or more years. It would not be surprising if Fermat knew something that is unknown today.

I usually dismiss claims that there is ancient knowledge which we have not rediscovered. For example, there is a myth about a process for hardening copper known 2000 years or more ago, which I consider bunk. On the subject of Fermat, I would not dismiss claims about his having yet to be rediscovered knowledge. I would not bet big money the other way either.

I believe that there are some mathematicians still trying to come up with a proof consistent with the knowledge available to Fermat. Nobody believes that the Fermat could have developed the modern proof. There are some very bright people who are not willing to sell him short by claiming that he did not have a different but valid proof.

[Good Dreams]

Originally posted by Guv


I believe that there are some mathematicians still trying to come up with a proof consistent with the knowledge available to Fermat. Nobody believes that the Fermat could have developed the modern proof. There are some very bright people who are not willing to sell him short by claiming that he did not have a different but valid proof.

I'm very pleased to know that. Once I have finished reading Fermat's Last Theorem I didn't like the end of it, no way. It started to be simple, but then the author simple ommited the more complex facts. It was more a romance than a scientific book. It brought some interesting thoughts. Clock Arythmethic(?) seemed interesting.

[Lord Orwell]

Originally posted by Guv

I usually dismiss claims that there is ancient knowledge which we have not rediscovered. For example, there is a myth about a process for hardening copper known 2000 years or more ago, which I consider bunk. .

I forget where it is, but there is this column made of steel that is 4,000 years old. The steel is so pure it has no rust on it. (pure steel won't rust)

[Starman]

Steel is an alloy of iron and carbon - how can you have pure steel ?

[Guv]

Every time I hear one of these myths it goes something like

My brother knows a guy who heard that . . . .

I forgot who told me, but . . . .

I do not know where I saw it or read it, but . . .

Never any information that can be verified.

[Guv]

Lord Orwell: The following 3 paragraphs are from various places in the article you mentioned in a previous post.

Friend[2] and Parr[3] indicate iron was known around 4000BC when it would have been pounded into shape. However, the source of this iron must have been celestial (meteorites) as the smelting process was probably not used on iron at that time. Further evidence for the celestial source of this iron is in the chemical analysis of artifacts. The high nickel content of these artifacts is typical of meteorites and the nickel content would also account for the lack of oxidation (rusting).

Whatever the actual origin of iron smelting it was a well known process by the end of the second millennium BC. Numerous quotations are found in the Homeric poems (circa 880 BC) referencing implements of iron. Herodotus makes reference to it in his "History" (446 BC) and Aristotle (350BC) attributes the sources of iron to mines in Elba and the Chalybian mines near Ambus[8].

It is suggested by Parr[18] that real production of steel began as early as 500 BC in India. This material was referred to as wootz. By Alexander's time the production of wootz was a well established two step process using the crucible method. Two methods could be used, conversion from a cast iron form or conversion from a wrought iron form.

They pretty clearly put steel smelting not much earlier than about 500BC, and iron smelting back around 1300 to 1000BC (at least that is the way I would interpret end of the second millennium BC).

The reference to iron from meteorites about 4000 BC is interesting. Note the mention of a high nickel content preventing rust. This is what might have given rise to the myth about an ancient steel column which did not rust.

I still claim that the steel column 4000 years old is a myth. The reference you provided does not contain any evidence to support that myth.

BTW: I thought that pure steel did rust, which is why they made so called stainless steel, which I believe (am not sure) has some constituents other than carbon and iron.

The article you cited talked about different percentages of carbon in steel, yet you refer to it as a compound. If it is a compound, what is its chemical formula? Steel is about 1% carbon, give or take a little, and definitely has less than 2-3% carbon. Fe99C1 or Fe200C3 would be about the formula for steel with 1 to 1.5% carbon. There is no way that such chemical formulae could be correct. Steel is not a compound, it is an alloy or mixture.

While not a chemist, the following does not seem right to me.

Probably the same way you have pure salt, which is a "alloy" of chlorine and sodium.
Note that salt is NOT a chemical bond, like water is.
(i mention this to prevent someone from making a fool of themselves)

If salt is an alloy, what is NaCl? If salt is an alloy rather than a compound, would could you make it with a different ratio of sodium and chlorine atoms than one for one? If it were an allow or misture, you could. It is my understanding that alloys are like mixtures, in that you can use different ratios of the constituents. Compounds have fixed ratios of the constituents because each molecule of the compound contains an integer number of atoms, and for a given set of elements, only certain combinations are possible. My periodic Table software refers to NaCL (Table Salt) as a compound, not an alloy. I have found this software to be an extremely reliable source of basic information on the elements and some basic chemistry. I do not know what you mean by “alloy” and “Chemical Bond” in this context. I never heard of salt being referred to as an alloy.

If this makes me a fool, it will not be the first time it happened to me. I consider NaCl (common table salt) to be a compound, not an alloy. When in solution, it is my understanding that it disassociates into sodium and chlorine ions, but on its own, it is a chemical compound, with the sodium and chlorine atoms bound by valence electrons. The periodic chart has them in just the right places for a one to one valence bond.

I will be amazed if some modern chemistry text says otherwise.

[Lord Orwell]

You will notice that i had "alloy" in quotes.

It is a compound, but so is steel. The difference is that the salt compound forms a crystal. The very crystal lattice will prevent variances in the mixture.
But let's assume the mixture is wrong: too much sodium, the sodium will quickly bond with hydrogen and oxygen. VERY quickly. A lump of sodium will explode if dropped in water.
Chlorine also will get removed before you get ahold of it in your shaker, since it is poisonous in any appreciable dosage.

[Starman]

That's the way they taught it to me Guv. All to do with adding up the right number of positive and negative valencies.

I suppose that the oxidisation resistive properties of these meteorites were what allowed them to reach the ground in sizes large enough to be useful. Any without this property would burn away quicker on the way down.

[Starman]

Lord Orwell,

Clourine is a gas at normal temperatures - it will drift away rather than having to be removed.

As I understand it these two very reactive elements will try to combine with anything near them, dislodging elements from compounds with weaker bonds where they can.
So common salt is as Guv said, a one-to-one mixture of Na and Cl. Steel can have any amount of impurities, compounds can contain iron - iron sulphate, iron clouride but steel is not a compound.

[THEROB]

It is true that in order to solve an equation - you must have the same number of unknowns as equations.

But how can Fermats last theorem (or conjecture) be proven when it has 4 unknowns and only one equation??

Rob

[Guv]

You are correct that one equations in 4 unknowns cannot have a unique solution. but consider the following equation.

x^2 + y^2 = z^2

It has an unbounded number of integer solutions. For example, (3, 4, 5), (5, 12, 13), (20, 21, 29), (39, 80, 89)

Fermat said that for an exponent greater than 2 there are no integer solutions.

There have been proofs for special cases for over one hundred years.

[THEROB]

Yes I heard that it can be proven for all of the odd numbers up to a 1000 using complex numbers - do you know where I can see one of these proofs?


Rob

[Pentax]

Does anyone know if it's true for the negative version?
I.e. doues x^n + y^n = z^n
have any solutions for n<-2? (or does it for n=-2?)

Have anyone ever tried to find out?
Cheers,
Pentax

[Guv]

Fermat's last theorem is from the mathematical discipline called Number theory, whihc is primarily (if not exclusively) concerned with theorems & conjectures about integers.

Negative powers are reciprocals of positive powers, and therefore not integers. People concerned with Fermat's Theorem just would not consider negative powers.

Without much analysis, I suspect that the theorem can easily be proved for negative powers.

1/x + 1/y = 1/z does not look as though it can be true for x, y, & z integers.

1/x + 1/y = (x + y)/x*y, does not look reducible to a form like 1/z.

If there are no integers satisfying the above, it only gets wors for.

1/x^n + 1/y^n = 1/z^n

[Guv]

I did not think that x and y were allowed to be the same.

Even if they are, it looks as though the following can never be if n is 2 or greater.

1/x^n + 1/y^n = 1/z^n

[Lord Orwell]

2 or greater? don't you mean 3?
these numbers all work (for example) as a b c for the ^ 2
20 15 12
15 20 12
40 30 24
30 40 24
60 45 36
45 60 36
80 60 48
60 80 48

[Guv]

Lord Orwell: Surprised by your (20, 15, 12) example. This is not the first time my intuition has been wrong.

I did not expect there to be any such examples. BTW: All (or most) of the examples you posted are really the same example, which does not detract from the fact that you found one example. This suggests that there are more non trivial examples.

When I say your examples are really the same example, consider that the Pythagorean triangles (3, 4, 5), (4, 3, 5), and (6, 8, 10) are not considered different Pythagorean triangles.

[Lord Orwell]

yeah, i noticed that myself after i posted the list.
I didn't sit down with a pen and paper and puzzle it out.
I cheated and wrote a program that generated the list

[Lord Orwell]

65 156 60

255 136 120

[Lior]

I made a simple program, which gave me all of the numbers between 1-100 which follow the Pitagoras rule: X^2 + Y^2 = Z^2

3 + 4 = 5
8 + 6 = 10
12 + 5 = 13
12 + 9 = 15
15 + 8 = 17
16 + 12 = 20
20 + 15 = 25
21 + 20 = 29
24 + 7 = 25
24 + 10 = 26
24 + 18 = 30
28 + 21 = 35
30 + 16 = 34
32 + 24 = 40
35 + 12 = 37
36 + 15 = 39
36 + 27 = 45
40 + 9 = 41
40 + 30 = 50
42 + 40 = 58
44 + 33 = 55
45 + 24 = 51
45 + 28 = 53
48 + 14 = 50
52 = 20 + 48
48 + 36 = 60
52 + 39 = 65
55 + 48 = 73
56 + 33 = 65
56 + 42 = 70
60 + 11 = 61
60 + 25 = 65
60 + 32 = 68
60 + 45 = 75
63 + 16 = 65
63 + 60 = 87
64 + 48 = 80
68 + 51 = 85
70 + 24 = 74
72 + 21 = 75
72 + 30 = 78
72 + 54 = 90
72 + 65 = 97
75 + 40 = 85
76 + 57 = 95
77 + 36 = 85
80 + 18 = 82
80 + 39 = 89
80 + 60 = 100
84 + 13 = 85
84 + 35 = 91
84 + 63 = 105
84 + 80 = 116
88 + 66 = 110
90 + 48 = 102
90 + 56 = 106
91 + 60 = 109
92 + 69 = 115
96 + 28 = 100
96 + 40 = 104
96 + 72 = 120
99 + 20 = 101
100 + 75 = 125

by the way, It has already been proven that this list is infinite.

[Destined Soul]

Don't forget Gauss! Not to shame all of the others, but he was insanely smart. It's too bad he didn't publish a lot of his findings. I wouldn't be surprised if he had solved Fermat's Last Theorum.

It's impressive what all of those guys did, though. You don't hear too much about that stuff today, however, as the topics are usually so localized to a specific field. It is pretty impressive that Wiles solved Fermat's last theorum.

If I can contribute even 1/10th of what they've accomplished, I'll be extremely content.

Destined.
(Physics Major & Math Minor)

[RAEsquivelC]

For x^2 + y^2 = z^2, we normally do not consider the “trivial” solutions, defined as those solutions which can be obtained by multiplying a “pure” solution by an integer.

We state that, if t is a factor of any two of x, y and z, it must also be a factor of the third:

If x = fs, and y = ft, then:
X^2 + y^2 = f^2(s^2 + t^2) and, therefore, z must = tu. So, all “t” may be divided out, obtaining, x=s, y=t, and z = u, relative primes (no common factor).

So, dispensing with all “trivial” solutions, we have:

x, y and z must be one of the following four:
1) All three are odd
1) x and y are odd and z is even
1) x is even, y and z are odd (Same for y even, x and z odd)
1) All are odd

1) If x and y are odd, z cannot be odd
This can be seen because, if x and y are odd, say:
x=2s+1, y=2t+1, then
x^2 + y^2 = (2s+1)^2 + (2t+1)^2
= 4s^2+4s+1 + 4t^2+4t+1
= 4(s^2+s + t^2+t) +2 = even, so z cannot be odd.

2) If x and y are odd, and z is even = 2u. Then, from 1),
4(s^2+s + t^2+t) + 2 = 4u^2. Dividing by 2,
2(s^2+s + t^2+t) +1 = 2u^2, which is false. So,
x and y cannot both be odd and z even

3) Say x is even, y and z are odd. Then
4s^2 + 4t^2+4t+1 = 4u^2+u+1, or
s^2 + t^2+t = u^2+u, which is possible.

4) All cannot be odd, because:
4s^2+4s+1 + 4t^2+4t+1 = 4(s^2+s + t^2+t) +2, which is even, so
z cannot also be odd.


Analyzing case 3):
x^2 + y^2 = z^2
x^2 = z^2 – y^2
= (z-y)(z+y)
Since x is even, = 2s:
4s^2 = (2u+1 – 2t-1)(2u+1 + 2t+1)
= (2u – 2t)(2u + 2t + 2)
4s^2 = 4(u – t)(u + t + 1) which, dividing by 4:
s^2 = (u-t)(u+t+1)
Letting u-t = Q, and u+t+1 = P:
s^2 = P*Q
It can be proven that P and Q are relative primes, so, the only solution possible is for P and Q to be squares, P = p^2, Q = q^2, giving:
s^2 = (p^2)(q^2), or
s = pq, from which,

x = 2s = 2pq

From
u + t + 1 = P = p^2, and
u – t = Q = q^2, which, adding, gives:

2u +1 = p^2 + q^2, or,

z = p^2 + q^2

And, subtracting the previous two equations gives:
2t + 1 = p^2-+ q^2, or,

y = p^2 – q^2

Now, x^2 + y^2 = z&2. Substituting the values found:
(2pq)^2 + (p^2-q^2)^2 = (p^2+q^2)2
4(p^2)(q^2) + p^4 – 2(p^2)(q^2) + q^4 = p^4 + 2(p^2)(q^2) + q^4
2(p^2)(q^2) + p^4 + q^4 ==p^4 + 2(p^2)(q^2) + q^4 Q.E.D.

So, with the values for x, y and z:

x = 2pq
y = p^2 - q^2
z = p^2 + q^2

it is easy to obtain all values desired that satisfy the equation,
x^2 + y^2 = z^2, noting only that p and q must be relative primes, with p > q and, for non-trivial solutions, one of p and q must be even, the other odd.

Examples:
p=2, q=1: x=4, y=3, z=5
p=3, q=2: x=12, y=5, z = 13
p=40, q=9: x=720, y=1519, z=1681

P.S. It is stated, in “Mathematics and the Imagination”, that the Egyptians knew of this relationship, and even used three large (in the 6- and 7-digit range) primes inscribed on a wall of a famous tomb, satisfying the x^2 + y^2 = z^2. WOW!

[Guv]

RAEsquivelC The following is nonsense.

P.S. It is stated, in “Mathematics and the Imagination”, that the Egyptians knew of this relationship, and even used three large (in the 6- and 7-digit range) primes inscribed on a wall of a famous tomb, satisfying the x^2 + y^2 = z^2. WOW!

One leg must be an even number, which cannot be a prime. I doubt that the book you cited made such an error.

[RAEsquivelC]

You are SO right! Of course, I erred! Thank you Guv, for keeping me honest.