The History of Fermat’s Last Theorem:
Fermat’s last theorem traces back to Greece in the 500’s BC with Pythagoras’ study
of numbers. In order to fully examine a special case of Fermat’s last theorem we must
trace its roots and understand the complexity of developments that led to a solution for this
case of Fermat’s last theorem.
Pythagoras was born on a Greek island, Samos, in 569 BC and died around 500 BC
in Metapontum, Lucania, Italy. Pythagoras traveled in Egypt learning math and various
other subjects such as astronomy and music. He was also known to be a healer, wrestler,
and an active politician. Politics drove Pythagoras from Samos to Crotona in what is now
Southern Italy. Pythagoras then moved to this Greek colony in southern Italy to escape his
dislike for the ruler of Samos. In Crotona he created a school based on a movement of
religious, political, and philosophical goals. The followers of his school were under strict
orders and maintained secrecy as they worked with Pythagoras to make new discoveries
and theories.
Pythagoras’ contribution to math was a remarkable theorem that was known to
other mathematicians, particularly some in Babylonia more than 1000 years earlier, but
had not been proven until Pythagoras investigated odd, even, prime, and square numbers.
After careful study, Pythagoras and his followers developed a basic understanding of
geometry and proved the Pythagorean theorem. The Pythagorean theorem states that the
square of the hypotenuse of a right triangle is equal to the sum of the squares of the other
two sides.
Pythagorean theorem: a2 + b2 = c2
A Pythagorean triple is a set of positive integers a, b, and c such that a right triangle
exists with legs a, b, and hypotenuse c. By the Pythagorean theorem this is equivalent to
finding positive integers a, b, c satisfying a² + b² = c².
The Pythagorean Triples Theorem states that any primitive (gcd(a,b,c)=1)
Pythagorean triple (a,b,c) of natural numbers such that a is odd and b is even is given by

a=p*q

b = (p2 - q2)/2

c = (p2 + q2)/2
for suitable odd numbers p and q with p>q, and with no common divisors. Since p and
q are odd, their squares will be odd, so their sums and differences will be even, leading
to positive integers for b and c. The proof that this works is quite simple:
Pythagoras was one of the great thinkers in the 500’s BC. His theories are still
used to day. The use of Pythagorean triples enabled later mathematicians to branch off and
make conclusions about the sum of powers of integers. Pythagoras did much more than
just prove the Pythagorean Theorem; Pythagoras and his followers contributed to music,
and astronomy, in addition to mathematics. Pythagoras believed in secrecy and
communalism which made distinguishing his work from the work of his followers is nearly
impossible. When joining Pythagoras's group, scholars had to remain silent for five years
before they could contribute to the group. Some of their discoveries were right, and some
were proven wrong in time (Calkins, Keith G. “Biographies of Mathematicians –
Pythagoras”).
Diophantine equations are polynomial equations with integral coefficients to which
only integral solutions are important. The study of Diophantine equations is one of the
central areas of number theory; one that Fermat looked into nearly fourteen centuries later.
In general, it is difficult to tell whether a given Diophantine equation is solvable. For
example, the Diophantine equation x2 - 94y2 = 1 is solvable, although the smallest solution
is x = 2,143,295 and y = 221,064. The Diophantine equation x2 - 94y2 = -1, however, has
no solutions (Calkins, Keith G. “Biographies of Mathematicians – Diophantus”).
Diophantus, the father of these equations was born around 200 A.D. and died
around 284 A.D. both in Alexandria, Egypt. His work during the middle of the third
century on the theory of numbers led to his book Arithmetica. Diophantine’s book
contains 130 problems giving “numerical solutions to determine equations with unique
solutions and indeterminate equations.” The method of solving the Diophantine equations
is known as Diophantine analysis. Diophantine did not deal with negative solutions
because negative numbers and zero were viewed with suspicion in his time thus a solution
to any Diophantine equation must be a positive integer (Edwards, 3). Diophantine did not
use sophisticated algebraic notation, however he did introduce symbolism that used an
abbreviation for the unknown.
It was in Diophantine’s Arithmetica that Fermat wrote his last theorem, without
proof. Near the eighth problem of the second book, which asks, “given a number which is
a square, write it as a sum of two other squares” Fermat’s not states:
“On the other hand, it is impossible for a cube to be written as a sum of
two cubes or a fourth power to be written as a sum of two fourth posers or,
in general, for any number which is a power greater than the second to be
written as a sum of two like powers” (Edwards 2).
Then Fermat wrote a simple statement which has become known as Fermat’s Last
Theorem.
"I have discovered a truly remarkable proof which this margin is too small
to contain."
This small quote, found by his son Samuel, is all Fermat had to say in regards to his
theorem that xn + yn = zn, has no integer solutions for x, y, and z when n > 2. This note is
believed to have been written around 1630.
Pierre de Fermat was born on August 17, 1601 in Beaumont-de Lomagne, France
and died on January 12, 1665. This French mathematician is known as the founder of
modern number theory. He discovered the principle of analytic geometry, is known as the
inventor of differential calculus, and, with a friend, invented the theory of probability.
Oddly enough, Fermat attended the University of Toulouse where he studied law
until the second half of the 1620’s. After studying law, he moved to Bordeaux, a custom
of his day. Here he began developing and sharing mathematical interests around 1629.
Fermat had not given up on law; from here he went to Orleans and received his degree in
civil law from the University. Returning to Toulouse, he was appointed to the lower
chamber of parliament in 1631 and rose through the ranks in the Criminal Courts. Fermat
lived out the rest of his life in Toulouse.
Fermat made many contributions to the world of mathematics, however his
influence was severely hampered by his refusal to have anything published. Fermat was a
disorganized man who left many of his works incomplete, most significantly his last
theorem. Fermat loved challenging other mathematicians to solving problems, and got in
to arguments over critiquing others’ works. Other mathematicians began to be annoyed by
the complexity of some of Fermat’s problems.
Fermat came up with many theorems and played a significant role in geometry and
calculus throughout his life. Fermat’s favorite study was numbers and number theory. He
worked mainly with prime numbers and discovered that every prime in the form 4n + 1 is
always shown as the sum of 2 squares. One of his theorems says that if p is a prime
number and a is any positive integer, then ap -a is divisible by p. Fermat's study of curves
and equations promoted him to generalize the equations for an ordinary parabola ay = x2
and for rectangular hyperbola xy = a2. He also generalized the Archimedean spiral r = aØ,
this in turn directed him to a rule of mathematical procedure, that was equivalent to
differentiation. He applied infinitesimal quantities to other problems such as calculation of
centers of gravity and finding the lengths of curves. Fermat was also one of the few
mathematicians who disproved Descartes idea that the precise determination of the length
of algebraic curves was impossible (Calkins, Keith G. “Biographies of Mathematicians –
Fermat”).
Fermat is best known for his "Last Theorem", which is quite ironic because the
proof has never been found. Some say that Fermat never formulated a proof or that the
proof was wrong. We agree with the scholars who have doubt in Fermat because the
actual proof took over 300 years to discover and is quite long and uses mathematics that
was not known till much later than Fermat. It seems impossible that Fermat could have
come up with the theorem without careful thought and reasoning to eliminate any doubt. It
seems almost as unlikely that Fermat had a different, much simpler, proof in mind that has
so far evaded discovery. We think that Fermat had a good guess about his theorem, but he
could not put into words why it works with a substantial proof. Fermat’s Last Theorem is
merely an assertion that whose proof remained a mystery, quite possibly because it was
never originally formulated.
Fermat’s last theorem:
xn + yn = zn, has no integer solutions for x, y, and z when n > 2
branches out from observations he made about the Pythagorean theorem. Fermat
claimed that when Pythagoras’ theorem was changed to read
an + bn = cn
the new equation could not be solved in integers for any value n greater than 2.
Diophantus’ book probed for solutions when n equals to 2, but Fermat was curious about
numbers greater than 2. This theorem was created by Fermat prior to his death in 1665 but
confused mathematicians for over 300 years.
Euler claimed to have a proof for the theorem on August 4, 1753 for the special
case of n = 3, however this proof contains a fallacy and the proof with n = 3 cannot be
credited to Euler.
Sophie Germain made a major step in proving Fermat’s last theorem by splitting it
in to two cases (O’Connor, J. J. and Robertson, E. F. “Fermat’s Last Theorem”). Sophie
was one of the few women in math in the early 1800’s. She was the first female to attend
sessions at the French Academy of Sciences. One of her major contributions to number
theory was a theorem that stated: if x, y, and z are integers, and x5 + y5 = z5 then either x, y,
or z has to be divisible by five. This proof, which she first described in a letter to Gauss,
became quite significant as it restricted the possible solutions of Fermat's last theorem
(“Sophie Germain”).
Case 1: None of x, y, z, is divisible by n.
Case 2: One and only one of x, y, z, is divisible by n.
Sophie proved Case 1 for all n less than 100 around 1823. Legendre extended her
methods to all numbers less than 197. Case 2 was solved by Dirichlet and presented to the
Paris Academy of Sciences in July 1825, with a complete proof given in September of that
year.
Dirichlet published a proof of Fermat’s Last Theorem for n = 14 in 1832. The case
n = 7 was solved by Lamé in 1839. On March 1, 1847, Lamé factored xn + yn = zn into
linear factors over the complex numbers and announced to the Paris Academy that he had
proved Fermat’s Last Theorem. Unfortunately unique factorization into primes was
needed for the complex numbers and it was doubted to be true at that time, although today
we have resolved this doubt. In fact, 2i = 2*i and is also (1+i)^2 and 2 = i^3*(1+i)^2 (a
complex integer is any m + in, where m and n are integers.
On March 15, 1847, Wantzel claimed to have proved the theorem with the
argument that: it is true for n = 2, n = 3, and n = 4, and one easily sees that the same
argument applies for n>4. This argument was hopeful, but was not sufficient enough to be
a proof.
Prizes were offered to anyone who could provide a solution to Fermat’s last
theorem, yet it remained unsolved for over a hundred more years.
Andrew Wiles, a British scholar at Princeton University claimed to have proved
Fermat’s theorem during a three day seminar given at the Isaac Newton Institute in
Cambridge, England from June 21-23, 1993. This remarkable discovery seemed
believable, until December 4, 1993 when Wiles made a statement that a number of
problems had emerged and needed to be resolved before the proof was complete. On
October 6, 1994, Wiles sent the revised proof to three colleagues to critique and simplify.
On the 25th two reprints of the proof were released, the first by Wiles and the second in
collaboration with a colleague Richard Taylor.
Taylor lectured at the British Mathematical Colloquium in Edinburgh in April
1995, giving the impression that no real doubts remained over Fermat’s Last Theorem.
In the last decade current mathematicians have simplified Wiles’ argument and it is
now generally accepted. Fermat’s original proof has not been found and most likely never
will. Some scholars believe that Fermat’s Last Theorem has succeeded in doing one thing:
annoying a bunch of mathematicians for 300 years. Fermat always loved challenging other
mathematicians, and with his mysterious theorem he was able to do so for years after his
death. Perhaps Fermat had this intention when he wrote his short comment in the margin
of Arithmetica.
Fermat’s Last Theorem:
Our goal is to prove that The Diophantine equation xn + yn = zn has no positive
integer solution if n is any multiple of 4.
a. Explain why proving that the equation x4 + y4 = z4 has no positive integer solutions
would imply the same conclusion for the equation
x4k + y4k = z4k
for any positive integer k.
If x4 + y4 = z4 has been proven to have no positive integer solutions, then x4k + y4k =
z4k, the same is true for any positive integer k, if k = 1, then the equation is x4k + y4k = z4k
would be x4 + y4 = z4 and since we’ve already stated that x4 + y4 = z4 has no positive
integer solutions, then x4k + y4k = z4k cannot have positive integer solutions because it does
not hold for all cases.
b. Explain why proving that the equation x4 + y4 = z2 has no positive integer solutions
would imply the same conclusion for the equation
x4 + y4 = z4
Let x4 + y4 = z2 be referred to p, and let x4 + y4 = z4 be referred to q. No positive
integer solutions to p means that there is no positive integer solutions to q. So to prove
p=>q, we can prove that ~q=>~p. So ~q would be that x4 + y4 = z4 does have positive
solutions. Then ~p would also have a solution that p’s z = q’s z2. So because ~q=>~p,
p=>q.
Another way of saying this is to suppose q has positive integer solutions (a, b, c),
then p would have the positive integer solutions (a, b, c2). So if p has no positive integer
solutions, then q cannot have positive integer solutions because if q did have positive
integer solutions then p would have to as well.
c. If (x, y, z) is a solution of x4 + y4 =z2 consisting of positive integers x, y, and z with
no common factors, then by definition we call (x2, y2, z) a primitive Pythagorean
triple. Use the Pythagorean Triples Theorem to show that then the Diophantine
equation x4 + y4 = z2 has a solution (x’, y’, z’), where x’, y’, z’ are positive integers and
z’ < z.
The Pythagorean Triples Theorem for a primitive Pythagorean triple (X, Y, Z) is:
Every primitive Pythagorean Triple (X, Y, Z) satisfies:
X = uv, Y = (v2 – u2)/2, Z = (v2 + u2)/2 with u, v odd integers and v > u.
Since we are saying (x2, y2, z) is a primitive Pythagorean triple, then it would stand
to reason that (x’, y’, z’) is a primitive Pythagorean triple since they are both solutions to
the same question. So for any odd integer p and q with q > p, the solution would be
x2 = (pq) ,
y2 = ((q2 – p2)/2) ,
z = (q2 + p2)/2.
So for x2 = (pq), p and q must have a common factor, r. So let p = m2 * r, and q =
n2*r, with n > m since q > p. Since p and q share a common factor then so must y2 and z.
So r must equal 1 for x2, y2, z to be a primitive Pythagorean triple (which we already know
since it is stated in the problem).
With p = m2, q = n2, we can substitute this in for y2 and z. So x = mn, y2 = (n4 –
m4)/2 and z = (n4 + m4)/2. Since n and m are both odd, and n > m, then (n – m) > 1. So let
q’ = (n – m)2 and then p’ = (m – m)2 = 1. By substitution, x’ = √(n – m)2 * 1, so x’ = n –
m, y’ = (((n – m)4 – 1)/2), and z’ = (((n – m)4 + 1)/2).
This z’ < z, since z = (n4 + m4)/2 > (((n – m)4 + 1)/2).
d. Use the result of part c to prove that the equation x4 + y4 = z2 has no positive
integer solutions and thus conclude from parts a and b that the Diophantine equation
xn + yn = zn has no positive integer solutions if n is any multiple of 4.
Part c proves that if x4 + y4 = z2 has a positive integer solution, there is another such
solution where z > z’. By contradiction, part c ultimately proves that x4 + y4 = z2 has no
positive integer solutions. Since we have proven that x4 + y4 = z2 has no positive integer
solutions, by part b, we know that x4 + y4 = z4 has no positive integer solutions. And since
x4 + y4 = z4 has no positive integer solutions, by part a, this proves that x4k + y4k = z4k has
no positive integer solutions so we have proven this special case of Fermat’s Last Theorem
where xn + yn = zn has no positive solutions if n is any multiple of 4.
QED.
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