Pierre de Fermat
Joey Arnone
Pierre de Fermat’s life
Pierre de Fermat was born in 1607 in Beaumont-de-Lomagne, France. It was believed that his birth year was
1601 - however, it was discovered that it was actually his half-brother, also named Pierre, that was born in
this year. Fermat grew up with three other siblings, having two sisters and one brother. His father was both
a successful merchant and an appointed second consul to his hometown, and his mother was a parliamentary
noblesse de robe, in which she held a high rank in state office. Clearly, Fermat was born with a substantial
amount of money. This proved much to his advantage, as he was given the opportunities to attend expensive
and prestigious schools. As an adolescent, he attended the Les Cordeliers convent (as he was a Catholic) for
secondary, or high school. He was very interested in languages, and by adulthood became fluent in Greek,
Latin, Italian, Spanish and Occitan (a Romance language that is spoken in Southern France).
A portrait of Pierre de Fermat
He is known as one of the worlds most influential mathematicians, and for this reason it is slightly ironic that he
studied law at the University of Orleans. He began his studies in 1623 and graduated a mere three years later
as an 18 year old. In his young adulthood, Fermat took a strong liking to not only law, but also that of art and
science. When Fermat moved to Bordeaux in the late 1620s, right after graduating from University of Orleans,
he began making mathematical research of importance. He solved conjectures left by mathematicians from
early decades and made significant advances in the fields of algebra and geometry through this. After Fermat’s
father passed away in 1630, he inherited a great deal of money, and used a portion of that money to obtain a
senior position in the High Court of Toulouse. This was the profession that he carried out until the end of his
life. He married his wife, Louise de Long, only a year later, and they had eight children together, five of whom
survived into adulthood. Throughout his life, he collaborated with many different mathematicians and solved
many different mathematical proofd, producing invaluable work, including the creation of probability theory
with Blaise Pascal and the development of prime number theories that came as a result of collaboration with
Marin Mersenne. It was around this time that he developed his most famous work: Fermat’s Last Theorem.
Soon after creating his last theorem, he received a large promotion in his profession as a lawyer. In 1638,
he was granted a position as a lawyer in Toulouse’s criminal court. Soon afterwards, he was also granted to
work in the Grand Chamber, which was the highest honor in law work at the time, as he was able to deal
with largely political and international affairs. During his lifetime as a mathematician, he continuously kept
climbing the career ladder as a professional in the field of law. A decade after being selected for work in the
Grand Chamber, he became Parlement’s chief spokesman, acting as a speaker on behalf of official politicians
in power. He worked mainly as a spokesman for Pierre Seguier, who was the chancellor of France. Through
various obscure writings from his time appointed, it can actually be implied that his work in being a chief
spokesman was not his best.
In his later adulthood, a plague gripped the town of Toulouse, where Fermat resided, and he had been unlucky
enough to contract it. However, he was able to pull through and survive. He continued making mathematical
work that was considered influential up until his death. He died in 1665 in Castres, France.
Pierre de Fermat’s mathematical works
In his years at the university, he would make conjectures based from uncompleted work by famed mathematicians, and one of the pieces he analyzed was Plane Loci by Apollonius. In going over this work, Fermat
discovered that loci could be analyzed through usage of coordinates, in which case both algebra and geometry
is needed. This work alone has proved invaluable to the field of analytic geometry.
In the 1630s, Fermats life was preoccupied with many other matters besides mathematics - he became a
lawyer/government official and also started a family during this time. However, he still worked on developing
numerous theories, and it was at this point where he was at his most fruitious period. Fermats research began
leading him to analyzing curves and equations related to curves.
He extended the equations of parabolas
ay = x2
and hyperbolas
xy = a2
and was able to generalize these equations into the form
a( n − 1)y = xn
Curve estimates are then attained from using this formula, and they are now known as the parabolas and
hyperbolas of Fermat. The equations (and curves obtained from these equations) that he generalized for both
this and the Archimedian spiral became used for his discovery of what we now know as basic differentiation.
Most of his life works (coordinate algebra and geometry) came to a summation on this theory, as it deals with
locating maxima, minima, and inflection points, along with finding tangents to the curve. This work is known
as finding ”adequlities”, and it was first published in his treatise ”Methodus ad disquirendam maximam et
minimam”, or ”Methods for distinguishing maximums and minimums”, published in the 1630’s.
Soon after this, he made another important discovery. He was able to obtain a summation of the area bounded
by a curve through his earlier equations and as of consequence discovered the early form of what we now know as
integration. Given that Fermat delivered such important work dealing with derivation and integration decades
before Newton and Leibniz, one could possibly believe that it was Fermat himself who founded calculus.
2
However, Fermat’s examples only worked in specific cases, and no general solutions were ever worked out
with major success. Nonetheless, Fermat’s discoveries were highly influential in Newton’s work that led to the
founding of calculus, making Fermat possibly the biggest influence in its discovery. Newton himself wrote that
his work for developing derivation was influenced by ”Monsieur Fermat’s method of drawing tangents.”
Fermat also contributed to physics at this time, as his work on the law of refraction is still seen as influential
to this day. According to [Fermat’s] principle, if a ray of light passes from a point A to another point B, being
reflected and refracted (refracted; that is, bent, as in passing from air to water, or through a jelly of variable
density) in any manner during the passa ge, the path which it must take can be calculated- all its twistings
and turnings due to the refraction, and all its dodgings back and forth due to reflections- from the single
requirement that the time taken to pass from A to B shall be an extreme. Fermat believed that light passes
through denser material more quickly, which in essence is a justification of Snell’s law, which repeats basically
the same statement. He used his own theories on maxima and minima to help prove this - it later became
known as Fermat’s Principle. However, Fermats findings weren’t entirely true, as some ideas conflicted with
Aristotlean theory that nature (in this case, light) chooses the shortest path. The essence of what Fermat’s
Principle now states is that the path that a ray of light takes is the one in which it takes the least amount of
time.
Fermat began finding more success when he started his work on number theory. It was Fermat that discovered
the idea that prime numbers are numbers that can only be divisible by 1 and themselves.
He found an equation:
2k + 1
With this equation, Fermat objected, one can find many prime numbers - the numbers generated from this
equation are now known as Fermat numbers. Even with his discoveries, he still had a few shortcomings in the
mathematics he used. Euler proved later that one of the numbers generated from the Fermat number equation
actually has factors other than 1 and itself, proving Fermats number theory not entirely true. The level of
research that Fermat underwent to produce an equation like this, however, is astounding, as he studied not
only the properties of prime numbers, but also perfect numbers (positive integers that are equal to the sum of
their proper divisors), amicable numbers (two different numbers that are so alike that the sum of their proper
divisors is equal to the other number) and Pell’s equation
x2 − ny 2 = 1
Fermat learned from his mistakes while working on his prime number theory and used the knowledge he
obtained for the development of another theory on numbers. He soon discovered that every single prime
number that is divisible by 4 and leaves a remainder of 1 when this division occurs
n = 1, mod4
can be expressed as a sum of two square numbers.
p = x2 + y 2
His findings became known as the Two Square Theorem. An even more influential theorem that he responsible
for is his Little Theorem. In this, he states that if a and p are two numbers, and p is a prime number not
divisible by a, then multiplying a by itself p-1 times and thereafter dividing by p will leave a remainder of
1. This work is still highly influential today, as it is used in codes for credit card transactions, among other
things. Fermat to this day is seen as the father of modern number theory.
3
Fermat still delved into mathematical works on prime numbers after all of his years of working on them. He
discovered a group of numbers that are essentially the number 2 to the power of a power of 2, with 1 added on
to the equation. It was noted that all of these numbers were prime, and they later became known as Fermat’s
primes. To this day, there are still only 5 known Fermat primes.
Fermat’s primes, of which only five are completely known today.
All of Fermats life work led to the development of his most famous (and controversial) work - Fermats Last
Theorem.
This states that the equation
xn + y n = z n
with x, y, z, and n being positive integers, cannot be solved if n is greater than 2.
This equation was first published in the margins of ”Arithmetica” by Diophantus in the year 1637. The full
proof was not published, according to Fermat, due to there not being enough room to fit it. He wrote the
following: ”It is impossible to write a cube as a sum of two cubes, a fourth power as a sum of fourth powers,
and, in general, any power beyond the second sum as a sum of two similar powers. For this, I have discovered
a truly remarkable proof which this margin is too small to contain.”
Fermat’s publishing of his statement on his last theorem in a 1670 edition of Diophantus’ ”Arithmetica”
4
Interestingly enough, so many proofs for this theorem that were created by countless mathematicians were
proven wrong that it now holds a record in the Guinness Book of World Records as the ”most difficult math
problem”. The full theorem was not fully proved until 1994 when Andrew Wiles, a British mathematician at
Oxford University, finally accomplished this feat. It is still seen as a highly complex mathematical theorem
with much history and significance behind it.
Andrew Wiles, the British mathematician who was able to solve Fermat’s Last Theorem.
It is noticeable that after being struck with the plague in the beginning of the 1650’s, his work output slowed
down quite a bit. However, it began to pick up in a mere few years, proving that after Fermat’s Last Theorem,
he was still able to produce work that had a significant degree of influence. In 1654, he worked closely in
association with Blaise Pascal, and was able to solve complex mathematical equations regarding gambling
that led to the development of probability theory. In 1657, he was able to find a way in which to solve Pell’s
equation (it is interesting to note that Pell’s equation had been solved nearly a half a millenium before in
India). According to other French mathematicians at the time, it was known that Fermat found the smallest
value for ”N” up to 150 in the equation
x2 − ny 2 = 1
Supposedly, he was close friends with British mathmatician John Wallis, who was also one of the mathematicians responsible for the developing the building blocks that would prove to be the foundation of calculus, and
he encouraged Wallis to find a solution for ”N” from 151 to 313. After all of these years, Fermat proved that
he still had the will to use math as a means of recreation. His level of influence with regards to Pell’s equation
was so signfiicant that now it is sometimes referred to as the ”Pell-Fermat equation”.
Collaboration with other scholars
Fermat frequently socialized with important mathematicians, among them Marin Mersenne and Jean de Beaugrand. They all formed a solid web of communication with each other, doing things like holding math challenges
and solving each others theories.
From Fermat’s days of being at the University of Orleans, he developed a friendship with Pierre de Carcavi,
sending him the details of his mathematical works from time to time. In 1629, he began seriously working
with Beaugrand, and together they worked on the development of what is now known as finding maxima and
minima on a graph. This work was shared with Etienne dEspagnet, with whom Fermat was in contact with
due to their sharing of mathematical interests. Espagnet inherited a library of books on mathematics, and
this is most likely the reason for their close correspondence. After this, Fermat became increasingly influenced
by the workings of Francois Viete, a mathematician whose work was heavily focused on geometry and algebra.
5
Ironically enough, Descartes himself was an admirer of Viete, and one can imagine the effect that Viete has
had on both Descartes and Fermats work. Viete’s work, along with the work of Archimedes, became very
influential in Fermat’s work on maxima and minima, which he discovered at the age of 21.
Marin Mersenne was also an important person in the development of Fermat’s mathematical career. In 1636,
Mersenne proposed that mathematicians should be more open in communicating with each other and discussing
their ideas, so that they would be more able to help each other and further the development of mathematics.
He sent Fermat a letter requesting more information about his work, and Fermat replied almost immediately.
After receiving such an immediate response, Mersenne was quoted as saying he knew he was dealing with a
great mathematician. It is very possible that collaborating with Mersenne has influenced Fermat’s work on
prime numbers.
In 1654, Blaise Pascal wrote to Fermat detailing a set of mathematical gambling problems, as Pascal was very
interested in gambling himself. Through complex mathematical calculations, Fermat was able to solve all of
them efficiently. The mathematics that he used is what we now know as probability. Both Fermat and Pascal
are now seen as the leading figures behind probability theory.
Historical events that marked Fermat’s life.
When Pierre de Fermat was an infant, King Henry IV was assassinated by a Roman Catholic fanatic named
Francois Ravaillac, as there were growing tensions between Roman Catholics and Protestants in France. Soon
after, Louis XIII was made King of France, with Marie de’ Medici ruling as regent over the country. King
Louis XIII’s reign led to French involvement into the Thirty Years War, and subsequently the Franco-Spanish
War.
During Fermat’s lifetime, Rene Descartes published two works that are now seen as very influential. The first
of these are ”Discours de la methode”, or ”Discourse on the Method”, which was published in 1637. This book
is a treatise which is not mathematically driven as much as it is philosophically. Descartes displays many of
his religious beliefs and ideas regarding the human soul. It is in this book that the famous saying ”I think,
therefore I am” comes from. The second of these books is ”Meditations de prima philosophia”, or Meditations
on First Philosophy, which was published in 1641. It is seen as his most popular work that he has ever written.
The subtitle of the book is ”In which the existence of God and the immortality of the soul are demonstrated”.
It is yet another religious book which addresses the questions that many religious skeptics have and reveals
the proof of God’s existence.
6
A first edition of Descartes’ ”Meditations de prima philosophia”, which is considered to be his most
influential and popular work.
Into the later years of Fermat’s adulthood, the Fronde (France’s civil war) occurred. It was a five year war
(1648-1653) which had two chronological parts to it. The first was the Fronde of the Parlement, which lasted
from 1648 to 1649. By this time, Louis XIV was already in power. Louis XIV’s chief minister, Cardinal de
Richelieu, enacted policies in the years before which lessened France’s judicial powers and gave more power to
the aristocracy. The Parlement made a detailed reform list for the aristocracy. This enraged the aristocracy,
and as a result, two parlementaires were arrested. This led to an uproar in Paris and the two parlementaires
were released soon afterwards. However, this did not end the conflict. The Parlement stood steadfast in their
desire for reform and, as a result, a resolution was reached. The Peace of Rueil treaty was signed a few months
later and a compromise between the aristocracy and Parlement was initiated.
The second part of this war was the Fronde of the Princes, which lasted from 1649 to 1653. Soon after the first
Fronde was over, members of the aristocracy began opposing Jules Cardinal Mazarin, an Italian chief minister
who had a significant level of influence in France and played a large role in the first Fronde on the aristocratic
side. This opposition was further orchestrated by The Great Conde, who was Louis XIII’s cousin, as he began
to rebel against the aristocracy. This opposition led to rebellion, and as a result of this, Mazarin was taken
out of power, with Conde becoming the new chief minister of France. However, through the workings of Anne
of Austria (Louis XIV’s mother), Conde was eventually exiled from France. The war ended in 1653 with there
still being discord between the aristocracy and Parlement.
In 1651, a plague occurred which took a toll primarily in the town of Toulouse, where Fermat lived. Fermat
himself contracted the plague, and in 1653 it was falsely reported that he had died of it.
Significant historical events around the world during Fermat’s life
When Fermat was a child, the Thirty Years War commenced, and it lasted from 1618 to 1648. Two years into
the war, the Huguenot Rebellion occurred in France. The Huguenot Protestants that lived in France were
protected by a treaty called the Edict of Nantes, which gave them rights and a means of civil unity. This edict
was signed by King Henry IV, a Roman Catholic, and it was successful for the entirety of his reign. However,
7
when King Henry IV was assassinated and Louis XIII took over, the principles found in the treaty began
falling apart. Under the direction of him and his mother, Marie de’ Medici, who were both Roman Catholics,
Protestants began undergoing persecution and hostility. They quickly responded to this persecution by forming
their own military structures and cooperating with contacts from other countries as a means of gaining their
own power. In doing this, the Huguenot Protestants led a revolt against the French monarchy. Ultimately, the
Treaty of Montpellier was established, which ended the hostility between Protestants and Roman Catholics.
Due to events that were the result of the Thirty Years War, France began a war against Spain in 1635. Cardinal
Richelieu of France was in opposition to the level of influence that Spain had over France, and as a result, war
was declared. It is now known as the Franco-Spanish War, and it lasted for 24 years, when the Peace of the
Pyrenees treaty was signed. As a result of this treaty, France was able to obtain territories that they had won
over, and they had to cut off ties with Portugal, whom they had a significant influence over.
A portrait depicting ”Les Batailles des Dunes”, or the Battle of the Dunes, a key battle during the
Franco-Spanish war in which France was victorious.
Significant mathematical progress during Fermat’s lifetime
In 1614, the logarithm was created by John Napier. It was conceived with complex mathematical calculations
and had invaluable influence in the fields of plane and spherical trigonometry, among other things. This
type of mathematics was very useful in helping astronomers do their work, reducing their need to use more
complex calculations. It was mainly influential in helping Newton with his own work during the 17th century.
Napier himself was a bit like Fermat in the sense that they both were considered ”amateur” mathematicians.
Napier only worked on mathematics as a hobby, yet left a legacy of highly influential work in his wake.
Initially, his calculations on logarithms were a bit flawed, and as a result he reworked them along with English
mathematician Henry Briggs. Other notable works of his is the popularization of the decimal point, along with
creating an easier means of lattice multiplication through ”Napier bones”, using numbered rods to calculate
products.
Marin Mersenne also produced significant mathematical work during this time. Perhaps his most memorable
work was that of the creation of Mersenne primes, which are prime numbers that are one less than a power of
8
two. It was considered pure mathematics, as it had no relativity towards practical use. However, nowadays it
is used for the creation of algorithms.
Fermats Plane Loci work was also discovered by Rene Descartes at around the same time. Fermats work on
analytic geometry was published after his lifetime (it was published in 1679 as Introduction to Plane and Solid
Loci), so it was Descartes that received the credit for the work. Even today we refer to most analytic geometry
as Cartesian geometry. There was a dispute between Descartes and Fermat about this issue. Secrecy among
other mathematicians was common around the 17th century, so naturally disputes like this would occur. As
for himself, Fermat largely communicated his mathematical studies through letters to friends, and he would
never give the proofs to any of his equations. However, the work that Descartes published during this time has
proved invaluable to many fields, including astronomy, as the orbit of planets was enabled to be plotted into
points on a graph. The work that Descartes did during his lifetime proved very influential in the development
of calculus.
Blaise Pascal worked in close association with Fermat on a number of things, including probability theory.
However, Pascal produced very memorable work on his own. Perhaps his most remembered work is his ”Traite
de triangle arithmetique”, or Treatise on the Arithmetical Triangle, which is now known as Pascal’s Triangle.
It presents binomial coefficients in a tabular form and was very innovative for its time. Another French
mathematician that was producing substantial, albeit not as influential, work was Girard Desaurges. The
work that he produced was directly related to the field of projective geometry, which analyzes different shapes
as they are projected onto a non-parallel plane. His theory was that there is a ”point of infinity” at which
parallels meet. For example, he proved that it was possible for two non-congruent triangle to have points at
which all of their respective edges meet at a point on a collinear line. The work that had the most influence
of his was 1639 work ”Brouillon project dune atteinte aux evenements des rencontres dun cone avec un plan”,
or ”Rough Draft of Attaining the Outcome of Intersecting a Cone with a Plane”, in which conic sections are
analyzed through a projective lens.
Important work which proved influential to the development of calculus occurred in the 1630’s, which was
the period where Fermat became very serious and involved in his mathematical ventures. An early form of
derivation was discussed by many mathematicians, along with Fermat himself, including Isaac Barrow, Blaise
Pascal, Rene Descartes and John Wallis in numerous works. This is when Fermat came up with his own
equations regarding derivates that stemmed from his study of hyperbolas and parabolas. Another important
work that was useful for the development of calculus was Italian mathematician Bonaventura Cavalieri’s work
on integration. He developed the theory that if by taking cross sections of two different objects with two
different shapes one finds that the cross sections add up to the same amount, then the two objects have the
same area. This general idea led to the development of what we now know as Cavalieri’s Principle.
Undoubtedly the most influential and important mathematical work which took place during Fermat’s lifetime
was the creation of ideas and principles that would eventually lead to the discovery of calculus. In order to
solve problems regarding physics in a more efficient manner, Isaac Newton began working on creating equations
that we now know as the basis of modern calculus, drawing from many influences such as Fermat, Cavalieri
and others. During Fermat’s lifetime, Newton was able to create the generalised binomial theorem, in which
powers of a binomial equation are expanded arithmetically. Newton would go on to develop theories that
would lead to the creation of calculus, as he was continuously discovering how to find both the slopes of curves
and the area under curves - however it was not until a few decades after Fermat’s death that he would create
this.
Connections between history and the development of mathematics
The Renaissance was still blooming during Fermat’s time, and it had a big influence on the mathematical
9
and scientific works of many brilliant minds, including himself. Science and mathematics were at their peak
of creativity across Europe during this time, and this period is now known as the Age of Reason. After
the Copernican Revolution in the 16th century, with the heliocentric model now being seen as a reality,
mathematical equations regarding the complexities of the solar system were being developed by Johannes
Kepler, Tycho Brahe and Galileo Galilei. These equations were influential in the development of 17th century
mathematics, as many influential equations that were developed were accessible to astronomers, including John
Napier’s equations regarding logarithms.
Remarks
Pierre de Fermat made some truly exceptional advances in the field of mathematics during his lifetime. The
one thing that is so exceptional about it all is that he was never considered to be a true mathematician - he
was seen as an amateur. Mathematics to him was merely a hobby, but his passion for it was so strong that
he worked at it for decades and decades, publishing innovative works and becoming a well-known name for
something other than his actual profession. Out of every mathematician that has produced influential work
in the 17th century, he is by far the most influential with regards to the development of calculus. The way he
set his work up for derivation and integration, the two building blocks of calculus, were so well developed that
all it needed was more generalized equations to fit every single case, which is what Newton and Leibniz would
do in the late 17th century.
References
1. https://www.britannica.com/biography/Pierre-de-Fermat
2. http://www.storyofmathematics.com/17th_fermat.html
3. http://www-history.mcs.st-and.ac.uk/Biographies/Fermat.html
4.http://mathsforeurope.digibel.be/pierredefermat.html
5. http://www-history.mcs.st-and.ac.uk/Biographies/Viete.html
10
6. https://www.math.rutgers.edu/~cherlin/History/Papers1999/chellani.html
7. http://www-history.mcs.st-and.ac.uk/HistTopics/Fermat’s_last_theorem.html
8. http://www.famousscientists.org/pierre-de-fermat/
9. https://www.britannica.com/biography/Henry-IV-king-of-France
10. https://www.britannica.com/topic/Huguenot-Wars
11. https://www.britannica.com/topic/Franco-Spanish-War
12. http://plato.stanford.edu/entries/descartes-works/
13. https://www.britannica.com/event/The-Fronde
14. http://www.storyofmathematics.com/17th.html
15. https://www.britannica.com/biography/Girard-Desargues
16. https://www.britannica.com/biography/Isaac-Newton
17. http://www.thefamouspeople.com/profiles/pierre-de-fermat-5067.php
11