Fermat's Last Theorem by Joseph Arnone

Fermat’s Last Theorem Joey Arnone History

Pierre de Fermat

Pierre de Fermat was an incredibly influential mathematician from the 17th century, producing innovative works that have led to the development of calculus shortly after his lifetime. He has made amazing contributions to many different fields of math, including probability, prime numbers, geometry and algebra. His most infamous, and controversial, work remains to be one that is named after him, known as Fermats Last Theorem. It was first conjectured in 1637, written in the margin of a copy of Arithmetica by Diophantus. The work published in Arithmetica turned out to be very influential in the development of Fermats Last Theorem.

In Arithmetica, Diophantus gives a means of producing two distinct numbers such that, when both squared, equal the square of a given rational number. In other words, when u and v are squared, a rational number can be found such that k2 = u2 + v2. This very equation is what sparked Fermats ideas for his famous theorem. Fermat came to the consensus that the equation an + bn = cn cannot exist for any integer n greater than 2. He related this theory in the margins of a copy of Arithmetica, stating that his proof would be too large to fit in the margins of the book, therefore excluding any actual proof of this theorem. In doing so, he would set off the most questioned and meticulously studied mathematical problem of the 20th century, being named in the Guinness Book of World Records as the most difficult math problem.

Soon after he released his theory, Fermat released a proof which provides an answer to Fermats Last Theorem for n=4, stating that there cannot be integers a and b such that a4 + b4 would be equal to c2. Interestingly enough, the main focus of his proof was not his theorem at all, as he was trying to show how a right triangle cannot have an area equal to a square, and the implications of this proof were able to be utilised for n=4 in Fermat’s Last Theorem. To obtain this proof, Fermat used proof by inﬁnite descent, which was a means of showing that solutions to a certain equation cannot be made possible by implying solutions that keep getting smaller and smaller in value.

Two proofs were later given for n=3 by Leonhard Euler in the year 1770. In one proof, he used an innovative way of solving using irrational numbers. However, he made a mistake of reasoning in this proof and it was not successful, unlike the other one. The successful proof was able to prove Fermat’s Last Theorem for n=3 by infinite descent, similar to Fermat’s own proof for n=4. Euler’s proof that used irrational numbers was still very influential for mathematicians such as Kummer, Gauss and Dirichlet, who produced influential work regarding Fermat’s Last Theorem themselves.

Another important mathematician that was a notable contributer in working on Fermat’s Last Theorem was Sophie Germain. In 1816, the Academy of Sciences in Paris held a competition for ﬁnding a proof that solves Fermat’s Last Theorem. Germain did not actually submit the work she produced, or even publish it for that matter, but would have made a sudden impact if she did. By the time Germain began her work on solving Fermat’s Last Theorem, there were only legitimate proofs for n=3 and n=4. No proof given was ever generalized, even for just a certain group of numbers. Her work was innovative, and after slight adaptations by mathematician Adrien-Marie Legendre in 1823, it gave a proof to Fermat’s Last Theorem for certain prime numbers such that they equal 2n +1.

Sophie Germain’s work on Fermat’s Last Theorem was both innovative and inﬂuential

The case for n=5 was immediately available due to Germain’s proof. Alternative proofs for n=5 were created throughout the year 1825 by mathematicians Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre. Over the course of the next century or so, many case-speciﬁc solutions were developed by many diﬀerent mathematicians around the world. However, it was these initial mathematical proofs that held the most impact over the work that was to come.

In 1847, mathematician Gabriel Lamé announced that he conceived a proof for Fermat’s Last Theorem at a conference. However, many mathematicians present at the conference were more reluctant to accept Lamé’s work as being completely accurate. Mathematician Ernst Kummer was able to fix what went wrong in Lamé’s proof, and as a result, gave a solution for Fermat’s Last Theorem for regular prime numbers, which will be discussed later on.

In 1922, a mathematician named Louis Mordell made a conjecture which implied that Fermat’s Last Theorem had a ﬁnite number of primitive number solutions if the given exponent is greater than two. This conjecture

2 came back half a century later, as the 1970’s brought an influx of mathematicians using geometry and surface analysis to prove Fermat’s Last Theorem. This is what led mathematician Gerd Faltings (whose specialist area was algebraic geometry) to bring back Mordell’s conjecture and apply it geometrically through a theorem. Falting’s theorem, proved in 1983, was able to prove Mordell’s conjecture, stating that for every genus (a donut- shaped object commonly used in math) greater than 1 over a field of rational numbers, there is a finite set of primitive solutions.

The development of the Taniyama-Shimura conjecture in 1955 was integral in the foundation of a generalized proof for Fermats Last Theorem. Through this conjecture, it was found that elliptic curves were actually Diophantine equations, y2 = Ax3 + Bx2 + Cx + D. It was Yves Hellegouarch who, in 1975, began making the connection between Fermat’s Last Theorem and the elliptic curves generated from the Taniyama-Shimura conjecture. He stated that if there is an equation with an odd prime number ”l” as an exponent and coeﬃcients of a, b and c such that al + bl = cl, then an algebraic curve will coexist in which y2 = x(x − al)(x + bl) (a standard elliptic curve will not exist).

German mathematician Gerhard Frey, in 1982, took this idea one step further by showing that the curves attained by this equation would not be modular. He provided a theoretical counterexample which would disprove Fermat’s Last Theorem, and wrote that the graph of the resulting equation would not cohere to the Taniyama-Shimura conjecture due to special properties of the curve - they are known as ”semi-stable elliptic curves”. Frey was not able to fully prove his theory, as he could not prove that the curve obtained was not modular, yet he presented his information at a conference in 1985. It was later published the following year. French mathematician Jean-Pierre Serre was a strong advocate for Freys conjecture, claiming that some of his own works support it. He too believed that the Taniyama-Shimura conjecture could validate Fermat’s Last Theorem, providing a proof around the same time. His proof was able to use a case of semi-stable elliptic curves to imply Fermat’s Last Theorem. Similar to Frey, Serre’s proof was not fully complete.

What was missing from each of these proofs is that neither of them were able to fully prove that the curves obtained from their equations were not modular. In 1986, Ken Ribet was able to prove the non-modularity of the curves, and in doing so, solved what was known as the ”epsilon conjecture”, or in other words the missing gaps that Frey and Serre were not able to ﬁll. Ribet’s solving of this conjecture made the full connection between the Taniyama-Shimura conjecture and Fermat’s Last Theorem. His work was published four years later.

It wasn’t until the 1980’s that the connection between the elliptic curves obtained from the Taniyama-Shimura conjecture and Fermat’s Last Theorem was fully made.

3 One roadblock that perplexed mathematicians was the seeming unability to prove the Taniyama-Shimura conjecture. One mathematician that was willing to accomplish this was Andrew Wiles, a professor at Oxford, whose specialist area was elliptic curves. Wiles’ work consisted of counting Galois representations and making a comparison with the number of modular forms obtained. In 1993, he ﬁnally completed his work and presented what he believed to be a proof of the Taniyama-Shimura conjecture for semi-stable elliptic curves. However, there was a slight error in his work which prevented what he had presented from being a complete proof. With the help of one of his students at Princeton named Richard Taylor (a respected mathematician himself), Wiles was able to prove the Taniyama-Shimura conjecture for semi-stable elliptic curves one year later, thus proving Fermat’s Last Theorem.

After over 300 years of research and questioning, it was Andrew Wiles who would ﬁnally ﬁnd a general solution for Fermat’s Last Theorem.

4 Mathematical Works / Proofs

Fermat ﬁrst wrote about his famous theorem in Arithmetica by Diophantus, whose proof he claimed ”this margin would be too narrow to contain.”

Pierre de Fermat’s proof for n=4

Suppose that there is a solution for a4 + b4 = c2 (or (a2)2 + (b2)2 = c2), where it can be assumed that a2, b2 and c are coprime numbers.

Due to the solution to Pythagorean Triples (which describe the three given side lengths of a right triangle), we know that there exist coprime numbers x,y such that: a2 = 2xy b2 = x2 − y2, with x being odd and y being even c = x2 + y2

Another Pythagorean triple from this can be attained, as x2 = b2 + y2

From this, we can see that there exist k,l in N as a result of the solution of the Pythagorean equation such that b = k2 − l2 y = 2kl x = k2 + l2

Then the equation a2 = 2xy can be written as a2 = 2(k2 + l2)(2kl), which can be further simpliﬁed to 4kl(k2 + l2)

This equation can be even further simpliﬁed to (a/2)2 = kl(k2 + l2)

Given that k,l and k2 + l2 are relatively prime, it can be known that they are both perfect squares by Fermat’s

5 proof of inﬁnite descent, so theoretically k = u2, l = v2, k2 + l2 = w2

This implies that (u2)2 + (v2)2 = w2, or u4 + v4 = w2

Then k2 + l2 = w2 < a2 < c2, which cannot possibly be true. Thus, Fermat gave the ﬁrst single case proof for his theorem. Leonhard Euler’s proof for n=3

It can be assumed that x, y and z are co-prime numbers.

For this proof to be valid, certain criteria must be met for numbers p and q, such that: 1. The greatest common divisor of p and q is 1 2. p, q have opposite parities (one is even and one is odd) 3. p, q are both positive 4. 2p(p2 + 3q2) is a cube

The greatest common divisor for (2p, p2 + 3q2) would have to be either 1 or 3, due to these following criteria: 1. If there is a prime f that divides 2p and p2 + 3q2, it could not be 2 as p and q have opposite parity and one of them is odd. 2. If f > 3, there will exist P, Q such that 2p = fP and p2 + 3q2 = Qf 3. f is not 2, so 2 must divide P, and there will exist H = (1/2)P, p = fH

From combining the two equations, this equation is obtained:

3q2 = Qf − p2 = Qf − f 2H2 = f(Q − fH2) f > 3, so it will not divide 3, but also divide q. It would also divide p. Then p, q would not be coprime, which is a contradiction of the statement that the greatest common divisor is either 1 or 3.

It can be further proved that if the greatest common divisor is 1 or 3, there will be smaller solutions obtained, which hint at the existence of even smaller solutions.

Thus Euler proved Fermat’s Last Theorem for n=3 using inﬁnite descent. Sophie Germain’s proof for 2p + 1 primes

Let p be a prime number and q an auxiliary prime which equals 2p+1 and fulﬁlls the criteria:

1. There are no consecutive remainders when p is raised to a certain number and divided by q

2. p cannot be a remainder when p is raised to a certain number and divided by q

If these criteria are met, then p2 must divide either x, y or z in the equation xp + yp = zp, which would be nearly impossible. Germain’s proof is interesting, as it does not directly state the impossibility of such cases, but points to the high degree of unlikeliness of its occurrence, as either x, y, or z would have to be a multiple of p. Germain’s proof for n=5

Suppose there is a solution for x5 + y5 = z5, where the product of the coeﬃcients will not equal 0.

Since it can be assumed that x, y, and z are coprime numbers, the greatest common divisor of (x, y, z) = 1.

6 It can also be assumed that only one of the integers are even, given that for the greatest common divisor of a set of three numbers to equal 1, two have to be odd. We can assume that x is even.

There exists z’ and x’, where z’ = -z and x’ = -x:

(−1)5(x0)5 + y5 = (−1)5(z0)5

(z0)5 will be added to both sides to obtain:

(−1)5(x0)5 + y5 + (z0)5 = 0

(x0)5 will then be added to both sides: y5 + (z0)5 = (x0)5

The equation z5 = x5 + y5, is then obtained, and it can be assumed the product of x, y, z equals 0.

It can then be proved that there can be no integer solution for n=5. Ernst Kummer’s work on Fermat’s Last Theorem for regular prime numbers

When Gabriel Lamé presented his supposed proof of Fermat’s Last Theorem, it was not well received. The work he presented was impressive, but had technical flaws that kept it from becoming a definite proof. He had factored xn + yn using complex numbers that, when raised to n, equal 1. These numbers are known as roots of unity, with notation ζ. The equation he obtained was as follows: xn + yn = (x + y)(x + ζy)(x + ζ2y)...(x + ζn−1y)

Gabriel Lamé believed that he had discovered a unique factorization into what are known as ideal primes, which can be described as a subset of numbers that share common properties of prime numbers. What Lamé did not know was that mathematician Ernst Kummer had already proved years before that the unique factorization spoken of in the proof was not valid for pth roots of unity of pth cyclotomic fields. Cyclotomic fields are number fields that adjoin primitive roots of unity to the field of rational numbers. This field is denoted as K = Q(ζp). In the weeks after Lamé’s presentation, Kummer was able to discover where Lamé went wrong with his proof. Kummer found a set of prime numbers which would be able to accomplish the unique factorization spoken of in the proof, and these are known as regular primes, which can be distinguished based on the class number of the pth cyclotomic field. (Class number can be thought of as a scalar quantity which details how close elements of a ring of integers to obtaining unique factorization.)

Kummer developed two diﬀerent proofs for this, and was able to solve Fermat’s Last Theorem for regular prime numbers. Andrew Wiles’ general proof for Fermat’s Last Theorem Andrew Wiles’ work for solving Fermat’s Last Theorem consisted largely of trying to prove that all elliptic curves had to be modular, and he carried this out by studying the proportion of the set of elliptic curves to the set of modular elliptic curves, as they had to be exactly the same in number. Wiles made this occur through counting Galois representations (vector spaces over a ﬁeld) of elliptic curves and matching them to modular forms. The equation can be denoted as follows:

Rn → Tn

7 To accomplish this, Wiles tried using a class number formula, which gives the class number of a ring (complex mathematical set) of numbers. A finite series of numbers (or curves, in this case) will be obtained from the usage of this formula. Wiles created this specific formula for obtaining this information through what is known as horizontal Iwasawa theory, in which extensions of number fields are obtained.

Wiles was not successful in using horizontal Iwasawa theory to obtain a class number formula, and therefore resorted to other methods. He used methods from mathematician Matthias Flach to develop a class number formula. He extended a certain method by Flach using ideas from mathematician Victor Kolyvagin during the creation of this class number formula to try and generalize his work for all elliptic curves, and this is exactly where his work fell short, as there was a ﬂaw in the mathematics he used in that portion.

Wiles’ full work on Fermat’s Last Theorem, presented in 1993, turned out to have flaws which kept it from becoming a legitimate proof. The following year, Wiles worked with Richard Taylor on correcting this mistake, and they discovered that the usage of Iwasawa theory as opposed to the work of Flach and Kolyvagin on the portion of the proof that was flawed led to a solution for all elliptic curves, effectively solving Fermat’s Last Theorem.

8 References

1. http://shayfam.com/David/flt/flt8.htm

2. https://www.britannica.com/topic/Fermats-last-theorem

3. Frey,Gerhard(1986),"LinksbetweenstableellipticcurvesandcertainDiophantineequations", An- nales Universitatis Saraviensis. Series Mathematicae, 1 (1): iv+40, ISSN 0933-8268, MR 853387

4. http://mathworld.wolfram.com/Taniyama-ShimuraConjecture.html

5. http://nautil.us/issue/24/error/how-maths-most-famous-proof-nearly-broke

6. http://fermatslasttheorem.blogspot.com/2005/08/sophies-proof.html

7. http://fermatslasttheorem.blogspot.com/2006/06/fermats-last-theorem-kummers-proof-for.html

8. http://www.math.mcgill.ca/darmon/courses/12-13/nt/projects/Colleen-Alkalay-Houlihan.pdf

9. http://fermatslasttheorem.blogspot.com/2006/08/ideal-numbers-class-number.html

10. http://fermatslasttheorem.blogspot.com/2005/05/fermats-last-theorem-proof-for-n3.html

11. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat%27s_last_theorem.html