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The Longest-Standing Math Problem

Article #70 • Written by Daniel Lew

Ever had a puzzle that looked easy but tortured you incessantly until you found a solution? Would you work on it obsessively for seven years in isolation? Andrew Wiles did just that to prove Fermat's Last Theorem.

Pierre de Fermat, a famous number theorist of the 17th century, rarely published his work – instead, he would often write comments in the margins of books. In one margin Fermat proposed that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2. However, rather than providing a proof, he only offered this taunting sentence: “I have discovered a truly remarkable proof which this margin is too small to contain.”

The proof for this simple conjecture was not solved for over 350 years and through the centuries became one of math's greatest puzzles.

Fermat's Last Theorem has entranced so many mathematicians due to its duality between simplicity and difficulty. It is easy to understand yet almost impossible to prove or disprove. That Fermat claimed to have a proof made it all the more intriguing. Mathematicians made some progress over the centuries in proving the correctness of the theorem for certain values of n – but this was hardly enough to prove that the theorem was correct for all values of n.

Then came along Andrew Wiles. As a child Wiles loved doing math problems. When he was ten he came across Fermat's Last Theorem which was, at the time, unsolved for 300 years. "It looked so simple, and yet all the great mathematicians in history couldn't solve it," said Wiles. "I had to solve it."

Wiles quickly became obsessed with solving the problem. Throughout his teenage and college years he worked on it, using his own methods and that of the mathematicians who had worked on it before him. However, when he became a research student he decided to put the problem aside. Wiles realized that current techniques could not solve the problem and that one could spend years without making any progress. Also, a proof of Fermat's Last Theorem would be completely useless to mathematics – it would not lead to anything useful for mathematicians. Instead, he went on to study elliptical curves at Cambridge.

His study of elliptical curves would prove useful, for in 1986 a new possibility was presented to Wiles. Ken Ribet linked Fermat's Last Theorem to another unsolved problem, the Taniyama-Shimura conjecture, which happened to be about elliptical curves. If one conjecture was true, both were – thus, if Wiles could prove the Taniyama-Shimura conjecture, he could prove Fermat's Last Theorem as well.

From that moment on he was determined to solve the riddle. He dropped all other projects he was working on and concentrated on the Taniyama-Shimura conjecture – in secrecy and isolation. “I realized that anything to do with Fermat's Last Theorem generates too much interest. You can't really focus yourself for years unless you have undivided concentration, which too many spectators would have destroyed.” His wife did not even know that he was working on the problem until he told her during their honeymoon.

Wiles worked on the problem alone for seven years. He devoted all of his time to working on the proof, the only exception being spending time with his family. He had a few breakthroughs but not a complete proof until one day in spring of 1993 in which he had the idea of examining the elliptical curves from the prime five instead of the prime three. Working feverishly (and forgetting to eat lunch), Wiles went to his wife that afternoon saying he had the proof.

Wiles introduced the proof in a series of three lectures which made no mention of Fermat's Last Theorem, but rather of elliptical curves. However, the audience realized by the end of the third lecture what Wiles was leading them towards. Once Wiles had finished his proof of the Taniyama-Shimura conjecture, he put Fermat's Last Theorem on the board then concluded saying, “I think I'll stop there.”

Wiles gained instant fame for having developed a solution to Fermat's Last Theorem. However, soon Nick Katz discovered that there was an error in a key section of his original proof. This setback proved difficult for Wiles to overcome, and none of the methods he tried could solve the error. He was about to give up when he re-examined his original (though discarded) method and found that there was actually a way to use it to resolve his mistake. “It was so indescribably beautiful,” said Wiles about the moment he solved the problem. “It was so simple and so elegant, and I just stared in disbelief for twenty minutes.” Thus, in 1994 the final proof of Fermat's Last Theorem was complete, weighing it at 200 pages, more complex than most people can understand.

The full riddle, however, is still not completely solved, for it remains unknown whether Fermat ever really had a brilliant proof to his conjecture. Fermat could not have thought of Wiles' proof – Wiles says that, "the techniques used in this proof just weren't around in Fermat's time." With the many mathematicians who had thought they'd solved it in the past, it is possible that Fermat deluded himself as well – but a simpler solution may still exist. Wiles, however, is content with his difficult proof - “I had this very rare privilege of being able to pursue in my adult life what had been my childhood dream.”

Article written by Daniel Lew, published on 06 December 2005. Daniel is a contributing editor for DamnInteresting.com.

Article design by Alan Bellows. Edited by Alan Bellows.
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28 Comments
Arcangel
Posted 07 December 2005 at 01:55 am

Ever had a puzzle that looked easy but tortured you incessantly until you found a solution?

Well actually .......... yes. Thanks for asking!

Back in March of 2000 a group of London publishers were offering the equivalent of 1.5 million Canadian dollars to anyone who could prove the theory that all even numbers greater than 4 were the sum of 2 prime numbers. Prime numbers for those not knowing is a number that is only divisible evenly by itself and one (examples are 1, 3, 5, 7, 11, 13, etc.). So for example, the even number 16 according to Goldbach would be the result of adding the two prime numbers 3 and 13. They gave participants until March 15, 2002 to come up with a solution. The publishers were not in the least bit worried about having to pay it out and they didn't.

This puzzle is known as Goldbach's first conjecture and was first raised by the Prussian mathematician, Christian Goldbach way back in the year 1742. It was discovered in correspondence between Christan Goldback and another mathematician by the name of Leonhard Euler from Switzerland. According to all accounts this theory has not been proven in the past 263 years but has been verified up to 4x10 to the 14 power by using an optimized segmented sieve and an efficient checking algorithm. Enough already!

It is easy to think of any even number and come up with 2 prime numbers that when added together total that even number. The problem though is trying to prove that it works for every even number imaginable. Now you know my dilemma so don't get me started on Goldbach's other conjecture! What is Goldbach's other conjecture you ask? Check it out for yourself at the link below: http://www.andrews.edu/~calkins/math/biograph/biogoldb.htm Now, sorry you asked?


mrBlah
Posted 05 January 2006 at 12:15 am

What about 1 not actually being prime, but special. A prime is divisible by itself and 1 only; two numbers. The number 1 is only divisible by itself; only one number.


Eric Leeson
Posted 24 January 2006 at 12:11 pm

I hate math.


vitualis
Posted 28 March 2006 at 10:12 pm

Read Simon Singh's "Fermat's Last Theorem" which goes through the history and, believe it or not, drama involved with the formulation of the eventual proof.

It's a very good read.

Michael Tam : vitualis' Medical Rants


Arcangel
Posted 10 April 2006 at 01:08 pm

Really surprised!!! I left the prime number 2 out of my first post thinking somebody would have picked up on it. I just came by now to check but nobody clued in on it, not even mrBlah.


grey matter
Posted 12 April 2006 at 08:39 am

what if that Pierre de Fermat guy was a publicity seeker and there is actually no solution????????


Random5
Posted 25 May 2006 at 01:00 am

Maybe they will solve Goldbach's first when someone figures out how to predict prime numbers, but almost certainly not before.


MAAT
Posted 15 June 2006 at 08:32 am

I was pondering the Mayan Tzolkin as l do sometimes and l had a thought about a simple numerology equation in my head, l was actually almost asleep, l suppose you would say meditation. Anyway at the end of it l came up with these numbers 1810, 1811, 1812, 1813, 1814, 1815, 1816, 1817 ,1818. They are repetitve in numerology. eg 1810

1+8=9 8+1=9 1+0=1
=991
9+9=18 9+1=10
=1810
This process just repeats. I havent found any other numbers that do this but havent spent much time on it. I have a hunch that there might be 13 more numbers like these. Do they have to do with genetics???
The Tzolkin is really a magnificent mathematical system, that needs great mathematicians working on it.
Good Luck, heres a start count the black marked days, then count the numbers in them. This is proof they were aware of our calender but chose to use theres.
Another piece of info for you the eye of Horus, is supposedly a mathematical code in Hieroglyphs. See http://www.aloha.net~hawmtn/horus.htm If you add all the fractions up and convert to decimal it comes to .99 Wasnt it the Egyptians who didnt know zero, or didnt they use it on purpose. Maybe everything in the world isnt 100% pure or completely destined, including us. The Illuminati certainly figured this out long ago, l'd say thats why the top of their pyramid is detatched(with one eye in it) hmmm sounds like a countries $ bill. Anyway we are maths(well 99%) and maths is us. I hope you all watch and study LOST/THEHANSOFOUNDATION.


MAAT
Posted 15 June 2006 at 08:37 am

Read Zecharia Sitchins books. Alot of ancient maths shown in Hieroglyphs, man made structures, god?? made structures and city placings etc. "What l thought was real, is now even more unreal, than what l thought to be unreal."


scamp
Posted 23 October 2006 at 03:00 pm

Im good at math, but im only 12.

So not as smart as you, golden oldies :)


Nov
Posted 27 October 2006 at 07:25 pm

Doing math makes me feel really retarted.


Cathryn
Posted 30 December 2006 at 12:27 pm

wow...that guys was like really smart


Cathryn
Posted 31 December 2006 at 12:45 am

hes a nerd too


JamesCuthbert
Posted 27 February 2007 at 06:38 am

Maths is not like riding a bike. i was ok at it when I was 15 now I'm 25 its almost all forgotten. Oh well...


ti83
Posted 30 April 2007 at 12:54 pm

Arcangel said: "Well actually ………. yes. Thanks for asking!

Back in March of 2000 a group of London publishers were offering the equivalent of 1.5 million Canadian dollars to anyone who could prove the theory that all even numbers greater than 4 were the sum of 2 prime numbers. Prime numbers for those not knowing is a number that is only divisible evenly by itself and one (examples are 1, 3, 5, 7, 11, 13, etc.). So for example, the even number 16 according to Goldbach would be the result of adding the two prime numbers 3 and 13. They gave participants until March 15, 2002 to come up with a solution. The publishers were not in the least bit worried about having to pay it out and they didn't.

This puzzle is known as Goldbach's first conjecture and was first raised by the Prussian mathematician, Christian Goldbach way back in the year 1742. It was discovered in correspondence between Christan Goldback and another mathematician by the name of Leonhard Euler from Switzerland. According to all accounts this theory has not been proven in the past 263 years but has been verified up to 4×10 to the 14 power by using an optimized segmented sieve and an efficient checking algorithm. Enough already!

It is easy to think of any even number and come up with 2 prime numbers that when added together total that even number. The problem though is trying to prove that it works for every even number imaginable. Now you know my dilemma so don't get me started on Goldbach's other conjecture! What is Goldbach's other conjecture you ask? Check it out for yourself at the link below: http://www.andrews.edu/~calkins/math/biograph/biogoldb.htm Now, sorry you asked?"

Actually, strangely enough, Goldbach's Conjecture has been proven to be independent of the axioms of ZFC (Zermelo-Frankel Set Theory, with the Axiom of Choice). That means that it is neither true, nor false. That's an interesting problem. Maybe you should write a DI article on that.


Ravage
Posted 22 September 2007 at 05:21 pm

Arcangel said: "Well actually ………. yes. Thanks for asking!

Back in March of 2000 a group of London publishers were offering the equivalent of 1.5 million Canadian dollars to anyone who could prove the theory that all even numbers greater than 4 were the sum of 2 prime numbers. Prime numbers for those not knowing is a number that is only divisible evenly by itself and one (examples are 1, 3, 5, 7, 11, 13, etc.). So for example, the even number 16 according to Goldbach would be the result of adding the two prime numbers 3 and 13. They gave participants until March 15, 2002 to come up with a solution. The publishers were not in the least bit worried about having to pay it out and they didn't.

This puzzle is known as Goldbach's first conjecture and was first raised by the Prussian mathematician, Christian Goldbach way back in the year 1742. It was discovered in correspondence between Christan Goldback and another mathematician by the name of Leonhard Euler from Switzerland. According to all accounts this theory has not been proven in the past 263 years but has been verified up to 4×10 to the 14 power by using an optimized segmented sieve and an efficient checking algorithm. Enough already!

It is easy to think of any even number and come up with 2 prime numbers that when added together total that even number. The problem though is trying to prove that it works for every even number imaginable. Now you know my dilemma so don't get me started on Goldbach's other conjecture! What is Goldbach's other conjecture you ask? Check it out for yourself at the link below: http://www.andrews.edu/~calkins/math/biograph/biogoldb.htm Now, sorry you asked?"

I also work on Goldbach's Conjecture. I would like to point out that the other conjecture is nothing more than a rewording of the 2 primes conjecture. If one has 3 primes, the sum of two of them are an even number by the primary conjecture. This means an odd (prime) and an even number equal an odd number. What this means is that we can take an arbitary prime (eg 3) and then using the primary conjecture create any odd number by selecting the correct two primes to create the requisite even number.

There is only one conjecture. The real answer to the conjecture is to prove that the prime numbers are clustered close enough together such that they can cover the binary partition of all even numbers less than double the largest odd prime.


Ravage
Posted 22 September 2007 at 05:30 pm

Random5 said: "Maybe they will solve Goldbach's first when someone figures out how to predict prime numbers, but almost certainly not before."

Predicting prime numbers isn't the issue, it's determining if an arbitary odd number is prime. Given a prime i the next prime will be within 2i. The problem is testing that i-1 range, if you already know the list of primes below i even that's not real hard.


Robert-H-Goretsky
Posted 24 September 2007 at 07:56 pm

Robert H. Goretsky says I always found the mathematics behind RSA encryption to be quite interesting. (Wikipedia Link ) What's crazy is how easy it is to generate a new public / private key pair, but how impossible it is to do the math 'backwards' to solve for the private key, given the public key... No computer in the world can solve these problems in our lifetime. Comment by Robert H. Goretsky of Hoboken, NJ


Ravage
Posted 27 September 2007 at 08:49 am

Robert-H-Goretsky said: "Robert H. Goretsky says I always found the mathematics behind RSA encryption to be quite interesting. (Wikipedia Link ) What's crazy is how easy it is to generate a new public / private key pair, but how impossible it is to do the math 'backwards' to solve for the private key, given the public key… No computer in the world can solve these problems in our lifetime. Comment by Robert H. Goretsky of Hoboken, NJ"

This depends on the size of the key. It used to be that 1024 or 2048 was considered strong. Now techniques are in use that allow cracking 2048 keys in reasonable amounts of time. At the current time a key length of less than 4096 is probably only moderately strong. And I expect that key length to crack in 2-3 years. There was an announcement today for example about two new quantum computing chips that were released. Once this technology comes to market about 2015 RSA, knapsack, and similar algorithms based on modulo and prime factoring will no longer be usable.


george9589
Posted 11 March 2008 at 03:28 am

Eric Leeson said: "I hate math."

I agree with you one hundred percent haha


catra7000
Posted 27 March 2008 at 02:17 pm

whenever i had a really horrible math problem that neither my mom or dad could figure out, i just said EFF IT and i'll get it wrong. cuz really, who cares? why waste the time and energy getting mad and stressed out by not getting the answer? when i decided to just pass on it, i felt so much more relieved and free


mabs239
Posted 15 September 2008 at 12:20 am

Very interesting column. But I donot know how to prove/derive the Euclid's formula :(


Benschop
Posted 21 October 2009 at 12:53 am

An elementary FLT proof (16 pgs, using semigroup Z(.) mod p^k, odd prime p)
was published in the Acta Mathematica of Univ. Bratislava (Nov.2005) see : http://pc2.iam.fmph.uniba.sk/amuc/_vol74n2.html (pp 169 - 184).
For some intro : http://home.claranet.nl/users/benschop/marg-abs.htm


mathematics101
Posted 17 February 2011 at 12:44 am

Ravage said: "Arcangel said: “Well actually ………. yes. Thanks for asking!

Back in March of 2000 a group of London publishers were offering the equivalent of 1.5 million Canadian dollars to anyone who could prove the theory that all even numbers greater than 4 were the sum of 2 prime numbers. Prime numbers for those not knowing is a number that is only divisible evenly by itself and one (examples are 1, 3, 5, 7, 11, 13, etc.). So for example, the even number 16 according to Goldbach would be the result of adding the two prime numbers 3 and 13. They gave participants until March 15, 2002 to come up with a solution. The publishers were not in the least bit worried about having to pay it out and they didn’t.
This puzzle is known as Goldbach’s first conjecture and was first raised by the Prussian mathematician, Christian Goldbach way back in the year 1742. It was discovered in correspondence between Christan Goldback and another mathematician by the name of Leonhard Euler from Switzerland. According to all accounts this theory has not been proven in the past 263 years but has been verified up to 4×10 to the 14 power by using an optimized segmented sieve and an efficient checking algorithm. Enough already!
It is easy to think of any even number and come up with 2 prime numbers that when added together total that even number. The problem though is trying to prove that it works for every even number imaginable. Now you know my dilemma so don’t get me started on Goldbach’s other conjecture! What is Goldbach’s other conjecture you ask? Check it out for yourself at the link below: http://www.andrews.edu/~calkins/math/biograph/biogoldb.htm Now, sorry you asked?”

I also work on Goldbach’s Conjecture. I would like to point out that the other conjecture is nothing more than a rewording of the 2 primes conjecture. If one has 3 primes, the sum of two of them are an even number by the primary conjecture. This means an odd (prime) and an even number equal an odd number. What this means is that we can take an arbitary prime (eg 3) and then using the primary conjecture create any odd number by selecting the correct two primes to create the requisite even number.
There is only one conjecture. The real answer to the conjecture is to prove that the prime numbers are clustered close enough together such that they can cover the binary partition of all even numbers less than double the largest odd prime."

i solved the problems you were looking at it was relativley easy


Benschop
Posted 26 July 2011 at 05:37 am

In http://de.arxiv.org/abs/math/0103091 you will find an 11-page elementary proof of GC.
Prove first for residues mod m_k (where m_k = \prod first k primes) that in the group G1(k)
of units mod m_k 'Goldbach-for-Residues' holds: each even unit is the sum of two prime units,
taking principle values of residues. Restrict summands to unit principle values u < q_k= [p_{k+1}]^2
(which all are necessarily successive primes because q_k is the smallest composite in G1(k) ),
and using Bertrand's Postulate ( p_{k+1} < 2p_k ) yield a proof of GC by induction over k,
with a proof by complete inspection for k=3 ( 4 < 2n < 30 ) as induction base.
---- Best regards, Nico Benschop.
PS: this proof is too simple to be published via the known peer-review process, as I learned...;-(


Benschop
Posted 26 July 2011 at 05:48 am

Correction: ‘Goldbach-for-Residues’ (mod m_k) : each even residue is the sum of two units.
and : Taking principle values of residues, restrict summands . . . .


Vestal
Posted 29 May 2014 at 05:02 am

mrBlah said: "What about 1 not actually being prime, but special. A prime is divisible by itself and 1 only; two numbers. The number 1 is only divisible by itself; only one number."

It took eight and a half years but here it is....finally.

That's why 1 is the loneliest number.


Cecil
Posted 14 January 2015 at 07:32 am

The algebraic solution ends in 5 places(_ _ _ _ _) answer, however the process is only 4 places( _ _ _ _ ) long. it takes a lifetime to solve.


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