Pierre De Fermat Ismail Tawil

Fermat’s life

Pierre de Fermat, a French mathematician born on the 17th of August, 1601 in Beaumont de Lomagne, France and passed away on the 12th of January, 1665 in Castres, France. He was the son of a successful leather merchant. While not much is known of his early life and education, there is evidence that his primary education might have been at a local Franciscan monastery; it is known that Fermat did attended the University of Toulouse and then later moved to Bordeaux in the second half of the 1620s. Fermat was ﬂuent in Greek, Latin, Spanish, and Italian. He also studied a little in philological problems and even some Latin poetry.

Fermat began his first critical mathematical research in Bordeaux. From Bordeaux Fermat went to the University of Orlans, where he studied law. He later received a Bachelor’s Degree in Civil Law. His mother and her side of the family were Pierre biggest influence for studying law and for deciding to work in Parliament. Fermat eventually married and had five children.

By 1631, Fermat was a lawyer and a government oﬃcial in Toulouse, as well as given the title of ”commissioner of requests.” In 1648, Fermat was promoted to a king’s councilor-ship in the parliament of Toulouse. The stature he now held entitled him to change his name from Pierre Fermat to Pierre de Fermat (”de” is the mark of nobility in France).

From his appointment in 1631, Fermat worked in the lower chamber of parliament but in 1638 he was appointed to a higher chamber, then in 1652 he was promoted to the highest level in the criminal court. Promotion was done mostly on seniority and when the plague struck the region in the early 1650s, many of the older men died. Fermat himself was aﬄicted with the plague and in 1653 his death was wrongly reported and was later corrected.

For the most part, math was a hobby to Fermat seeing as how he was a busy lawyer and did not let his love of math take up all of his time; however, he was at times preoccupied with his mathematical progress, so much so that he continued to cherish his mathematical friendship with a mathematician named Beaugrand, who he had previously done research with, even after he moved to Toulouse. Fermat also met another mathematician named Carcavi during his work in the government due to the fact that both of them were councilors in Toulouse; they both had a shared love for mathematics where Fermat discussed with Carcavi about his mathematical researches ad well as discoveries.

During his research in Bordeaux, Fermat studied the works of a French mathematician from the 16th century, named Franois Vite. Vites idea of algebra as an Analytic Art largely guided Fermat in his choice of the type of problems he had became interested in. Fermat viewed his own work as a sort of continuation to the Vitan tradition. From Vite, Fermat inherited the idea of Symbolic Algebra as a tool made for uniting the realms of both Geometry and Arithmetic, which would later inﬂuence his work in Number Theory. Algebraic equations had meaning in both ﬁelds, depending on whether the variables were line segments or numbers.

Moreover, Vites Theory of Equations shifted Fermat’s attention away from the solutions of certain formulas to the relationships between the solutions and the structures of their parent equations or between the solutions of one equation and those of another. Most of Fermats research headed toward Reduction Analysis, where any given problem could be reduced to another or identiﬁed with a certain class of problems for which the general solution was known. This Reduction Analysis could be reversed, in most cases, to act as a generator of families of solutions to problems.

Due to his love for language and literature and Vite’s inﬂuence on his young brilliant mind, Fermat was very concerned with the restoration of a number of lost texts and so he began to restore lost works such as Apollonius Plane Loci, Euclids Porisms, and Diophantus Arithmetica.

In terms of his mathematical work, there is speculation that Fermat never wanted any of his work to be published as he had considered math to be but a hobby. However, the few works that he did publish, he published them anonymously. Many of Fermat’s work was done by mail or correspondence; he would mail many other famous mathematicians his progress in the mathematical ﬁeld.

Fermat rarely provide his proofs in his correspondence with other mathematicians; usually stating the theorems, however, he had always managed to neglected the proofs of these such theorems. Due to his ”negligence”, many of his theorems were left without any proofs, which eventually led to much doubt as to his authenticity as a mathematician. In fact, his most famous work, which was dubbed ’Fermat’s Last Theorem’, stayed without a proof until, in 1993, a mathematician named Andrew J. Wiles provided the ﬁrst proof to this particular theorem.

Fermat received very little recognition as a mathematician during his life, and if not for the fact that other mathematicians saved his papers and letters, he may not have had the legacy that he has to this day. Together with Ren Descartes, a very distinguished French mathematician in the 17th century, Fermat was one of the two leading mathematicians of the ﬁrst half of the 17th century.

Following his death, his oldest son, Clment-Samuel inherited his law oﬃces in 1665 and then later decided to publish all of his fathers mathematical papers.

Fermat’s mathematical works

The Method of Maxima and Minima

An astronomer and mathematician by the name of Johannes Kepler, influenced the development of Fermat’s early mathematical works. Kepler began to observe the volumes for specific dimensions. Keplers observations determined that as a maximum value was approached from either side, the change in volume started to become increasingly small. Bearing this same concept in mind, Fermat chose to expand on this wokr by finding maximum or minimum values of any of the given functions.

Fermat understood that if a function f(x) had a maximum or minimum at x, then making e to be very small, it can be shown that the value of f(x − e) was almost equal to that of f(x). Therefore, this method involved what can be called a ”pseudo-equality,” in which Fermat carefully set:

f(x) = f(x − e)

However, before Fermat could ﬁnd the roots of this equation, which would enable him to ﬁnd the maximum and/or minimum of the function, he would need to correct the function’s ”pseudo-ness” by letting e ”assume the value of zero”.

2 Calculus and Curves

During his work on the properties of curves, Fermat helped in the development of the mathematical ﬁeld of calculus. As his studies of curves and their equations continued, Fermat noticed he could take the equation of an ordinary parabola (ay = x2), as well as an equation for a rectangular hyperbola (xy = a2), and generalize it to the follwowing form

an−1y = xn

The curves made by this equation are known as the parabolas/hyperbolas of Fermat (n is positive or negative). He similarly generalized the Archimedean spiral,

r = aQ

In the 1630s, he discovered an algorithm from these curves, that was similar to differentiation. This procedure let him find tangents of curves and locate the maximum, minimum, and inflection points of polynomials.

For any equation, Fermat’s way of finding the tangent at any given point was to find the subtangent for that specific point. Fermat found an equation for the areas bounded by curves through a summation process; this process has now been dubbed Integral Calculus. Such an equation is: Z A = xn dx = an+1/(n + 1)

It is not known whether or not Fermat noticed that the diﬀerentiation of xn, leading to

nan−1 was actually the inverse of integrating xn. Fermat took on problems involving even more general algebraic curves. Fermat applied his knowledge of infinitesimal quantities to many other problems, including the calculation of centers of gravity as well as finding the length of a curve. Fermat did not notice the idea of what is now known as the Fundamental Theorem of Calculus, however, his work in this area aided in the development of Differential Calculus.

Additionally, he contributed to the Law of Refraction1 by disagreeing with his colleague, a philoso- pher and amateur mathematician, Ren Descartes. Fermat had claimed that Descartes had incorrectly deduced his Law of Refraction because it was deep-seated in his assumptions. As a result, Descartes became irritated and attacked Fermat’s Method of Maxima and Minima. Fermat diﬀered with Carte- sian views concerning the Law of Refraction, which was published by Descartes in 1637. Descartes attempted to justify the Sine Law through the assumption that light travels faster the denser the two media involved in the refraction. Twenty years later, Fermat noticed that this seemed to conﬂict with the view of the Aristotelians where nature always chooses the shortest path.

Number Theory

1According to Fermat’s theory, if light passes from point A to point B, while being reﬂected and refracted in any manner during the passage, the path it takes can be calculated and the time it takes to pass from A to B is an extreme

3 One may deﬁne a prime number as any natural number greater than 1 which has a divisor (without remainder) of only 1 and the number itself. Fermat noticed that numbers such as 3, 5, 17, 257, 65537 all belonged to a single sequence and can be generated by one simple process:

3 = 2 + 1, 5 = 22 + 1, 17 = 24 + 1, 257 = 28 + 1, 65537 = 216 + 1

The sequence seen here is such an equation

2n ∞ 2 + 1 |n=1

Now if we wanted to see if one of these numbers is prime it would be a diﬃcult and lengthy process, unless we follow and understand Fermat who claimed that all numbers in the previous sequence are prime. However, Fermat was mistaken; he had just guessed and did not show any proof to his idea. However, a mathematician of the 18th century named Leonhard Euler, showed, for example, that 232 + 1 has 641 as a factor. So Fermat indeed was mistaken and we have yet to know whether there truly are any primes among the Fermat Numbers, for n > 4.

Although Fermat had made a mistake, through his work on numbers, he discovered that every prime number with the following form 4n + 1 could be expressed as the sum of two squares. However, Fermat as in almost all his mathematical works, left no written proofs of this theorem. He did however, write a letter to his friend, a mathematician named Carcavi. In this letter Fermat included the way in which he proved this theorem.

To start oﬀ, Fermat chose the number 5, which is the least of all primes that follows the earlier expression. According to Fermat, we must therefore infer by a Reductio Ad Absurdum that all numbers with such a form are sums of two squares. This caused much confusion because Fermat was not explicit as to how he proved this statement.

We may notice that Fermat was using a device which he called the method of ”Inﬁnite Descent”. Which is known as an inverted variation of reasoning by recurrence or by mathematical induction. An even more important result derived from this knowledge, is what is now known as Fermat’s Lesser Theorem. Fermat’s Lesser Theorem states that p is a prime number and a is any positive integer, then

(ap − a) | p

Once again, no proof was given for this theorem; however, in this case Leibniz, a 17th Century German mathematician as well as Euler, an 18th Century Swiss mathematician, provided the necessary proofs later on in time.

Fermat’s Last Theorem

Finally, Fermat’s last discovery will be brieﬂy discussed, dubbed ”Fermat’s Last Theorem”.

While he was reading Diophantus’ -a 3rd century Greek mathematician- book Arithmetica, Fermat made a note on the margin of the book that had remained unsolved until recent events where, as stated earlier, in 1993 Andrew Wiles proved this theorem. He had been reading the eighth problem in Diophantus’ book, which asked for the solution in rational numbers of the equation:

x2 + y2 = a2

4 He wrote that he had ﬁgured out how to solve the problem but due to lack of space, he never wrote down how he had solved the question. All that was discovered was a restatement of the original question by Fermat which stated: xn + yn = an, n > 2

However, Fermat did not live to prove what he had in mind. This theorem is still and always will be remembered as Fermat’s Last Theorem.

Collaboration with other scholars

In 1636 mathematician and friend of Fermat, Carcavi went to Paris as a librarian and met with Mersenne and his group of many other brilliant mathematician and the sort. After Carcavi began to talk about Fermat, Mersenne began to show interest in him when Carcavi described Fermat’s discoveries on free falling bodies, and due to his interest, Mersenne later wrote to Fermat.

Fermat replied to Mersenne’s letter and in addition to recounting to Mersenne the errors in which he believed that Galileo had made in his description of free fall, he told Mersenne about his work with spirals and his restoration of Apollonius’ text Plane Loci. This ﬁrst letter did also contain two problems on Maxima/Minima where Fermat had requested Mersenne to pass the given problem to his colleagues as well as the group of mathematicians that had formed around him. This was to be the typical style of Fermat’s letter:, he would challenge others to ﬁnd results which he had supposedly already known.

It would seem rather interesting that Fermat’s ﬁrst contact with the scientiﬁc community of Europe was about his study of free fall, due to the fact that Fermat had little to no interest in the physical applications of mathematics.

After a number of correspondence, mathematician Gilles de Roberval and Mersenne began to notice that many of Fermat’s problems were extremely diﬃcult and usually could not be solved using current techniques. They had asked of him to make his methods known, so Fermat later sent a method for determining Maxima/Minima and tangents to curved lines, as well as an explantion of his algebraic approach to geometry.

Fermat’s reputation as one of the leading mathematicians in the world came fast but most attempts to get any of his works published during his life had failed; mainly because he never really wanted to polish his work into a suitable form. However, some of his methods were published, for example, mathematcian Pierre Hrigone had added Fermat’s methods of Maxima/Minima to his major work: Cursus Mathematicus.

However, not everyone was dazzled by Fermat’s letters; Frenicle de Bessy became annoyed at Fermat’s problems mostly because they were impossible for him to understand due his weak understanding of mathematics at that point in time.

To add on to his interactions with many other brillaint minds, Fermat was sent a copy of Descartes’ text La Dioptrique by Beaugrand, however, Fermat paid little attention to it seeing as he was in the middle of a letter correspondence with Roberval and tienne Pascal where they had been discussing methods of integration and how one can use those methods to ﬁnd the centers of gravity in objects. Mersenne asked him personally to give his opinion on Descartes’ letter, which he ﬁnally did describing it as very vague and not focused; Fermat had claimed that Descartes had not correctly understood his

5 Law of Refraction.

Descartes became very furious at Fermat’s assumption. Descartes found any reason to attack Fermat’s work by viewing his work on Maxima/Minima and tangents as reducing the importance of his own work La Gomtrie. Roberval and tienne Pascal became involved in the argument and eventually so did Desargues; Descartes asked Desargues to act as a referee. Fermat did end up winning the argument and Descartes, after a while, began to accept Fermat’s work. However, such a clash with a mathematician as big as Descartes might as well have severely damaged Fermat’s reputation.

Fermat’s correspondences restarted in 1654 when Blaise Pascal, tienne Pascal’s son, sent him a letter asking for conﬁrmation about his ideas on probability. Their few letters paved the way for the soon to come Theory of Probability. Fermat however, tried many times throughout the correspondanec to change the topic from Probability to Number Theory. Pascal was not interested in Fermat’s Number Theory but Fermat still insisted and eventually sent him letters discussing Number Theory. However, Pascal showed Fermat that he certainly was not going to edit Fermat’s work, Fermat again gave up on B. Pascal and ended the correspondence.

Now invigorated, Fermat wanted to go further than ever with his challenging problems. His problems, however, did not pique too much interest as most mathematicians thought that his Number Theory was not an important topic. So Fermat sent more and more problems with a little more spice to them: like the sum of two cubes cannot be a cube or that there are exactly two integer solutions for x2 + 4 = y3 or the fact that the equation x2 + 2 = y3 has only one integer solution. He posed problems directly to the many English, Dutch, German mathematcians and all over Europe.

Around this time a student of Descartes’ was gathering Fermat’s correspondence for publication, so he turned to Fermat for help with what he hoped to call ”The Fermat-Descartes correspondence.” Fermat had been unsatisﬁed with Descartes’ description of the refraction of light so he settled on an idea which indeed did get the Sine Law of Refraction that Snell and Descartes had proposed.

Some assume that Fermat did not know how to proceed with many of the steps in his problems due to, again, his failure to write his methods and to show the proofs of his many works, which also eventually had made mathematicians lose interest in his work. It was not until Euler took up such problems that the missing steps were ﬁnally ﬁlled in.

One of his many problems, namely to ﬁnd all solutions of Nx2 + 1 = y2 for N not a square, was solved by Wallis and William Brouncker and they eventually developed Con- tinued Fractions in the solution.

Historical events that marked Fermat’s life.

In the 17th century, France was stable, for the most part, under the rule of King Henry IV. However, after his assassination in 1610, his son, Louis XIII succeeded him at the mere age of 9, making him an

6 easy target to the French Parliament. At this point France was pretty much being ruled by those that were a part of the Royal Court, which was the highest level of government in France at the time. As Louis XIII became of age, he began to assert himself more and in 1632, Louis XIII ordered the Royal Council to annul orders made by the Parliament while he was away campaigning because he believed that the orders by the Parliament were crossing the line of his authority.

Cardinal Richelieu dominated French history from 1624 until his death as Louis XIIIs chief minister. Richelieu was considered to be one of the greatest politicians in French history.

Many of his dealing with the stability and rise of the French powers were the Huguenots, a group who’s only purpose was the modernization of the French military and the eminent involvement of France in the Thirty Years Wars. In 1624, France eventually became involved with the Spanish in the Thirty Years War. While the central government was occupied with the war, the Huguenots took this opportunity to expand their power.

By 1630, Richelieu was in control of the Royal Court, though Louis XIII always insisted on the ﬁnal say in many decisions. Anyone who was to be in a seat of administration was to be hand picked by Richelieu himself, and many of them were chosen for their ability rather than their family background. As a result, the status of seniority was excluded from The Royal Court.

The biggest conspiracy Richelieu had to deal with was the Chalais Conspiracy in 1626. This involved a group of both low ranking nobles as well as high ranking nobles. Their plan was to assassinate Richelieu, get rid of Louis XIII and then share power amongst themselves; however, the plot failed due to Richelieu’s development of new a security information system. Richelieu had a titan’s grip on the nobles after this conspiracy was brought to light and eventually dealt with.

Richelieu only desire was to see France as a major European power. The overseas power France had was minimal and France had no colonial power at all, therefore, all ﬁnances had to come from France itself. Richelieu had to reform taxes, allowing only loyal regions a choice in their taxes while the taxes of the disloyal were forced. The high tax demands on the poor eventually began to take its toll and in 1636, a rebellion broke out in a place called Angoulme and yet again in 1639 another revolt took place in the town known as Normandy. Despite a number of outrages and pressure from the French people, Richelieu maintained his ﬁscal policies.

Richelieu deduced that a major European power must have a navy in order to survive and to protect any expanding trade so in 1629, Richelieu had decided that France truly did need a proper navy, so he began to expand and by 1636, France had a navy of nearly 40 ships. The navy was meant for both protection as well as encouragement of overseas trade now that shipping could be protected.

In 1628, to promote the establishment of colonies overseas, Richelieu created the Company of New France that encouraged settlement in French Canada. Richelieu had to use some of the money from his ﬁnancial policies to modernize France’s the army.

By his death, Richelieu had done what he always intended on doing: making France a serious player in European aﬀairs, a nation to be reckoned with.

Louis XIII trained Louis XIV to understand as much as he could about the Courts and the nobles. Louis XIV took the throne by the age of 5. He had a bit of an ego, he called himself the Sun King. Louis XIV took his role as king very seriously and tried his utmost in everything he did. During his

7 reign, Louis had a number of women present in his life; he had married at least twice and had 3-4 mistresses as well.

Signiﬁcant historical events around the world during Fermat’s life

There were many events that had taken place during 17th Century. While some of them beneﬁcial, other were not at all a help to the world at that time.

The Great Plague of London, an epidemic of plague that hit London, starting in late 1664 and ﬁnally coming to a stop in early 1666, it killed around 75,000 to 100,000 people out of a total population of only about 450,000 at the time.

Following not far after was the Great Fire of London which happened in 1666. The ﬁre destroyed a huge part of the London, which included 87 parish churches, and about 13,000 houses. The ﬁre had been accidentally started in the house of a baker near the London Bridge. The Fire is commemorated by The Monument, a column erected in the 1670s near the bakery where it had all started.

The Anglo-Dutch Wars were a trilogy of naval wars between the Dutch and England due to England’s attempt at barring trade from the Dutch. During the Second War England was pushed out by France who later joined the Dutch.

The Thirty Year War between the Catholics and the Protestants devastated most of Europe, but most of all: Germany, where it broke up into over thirty diﬀerent states.

Two civil wars happened in England, The English Civil War where Parliament took power and then the Glorious Revolution where power was moved from the Monarch to the Parliament.

The Americas began to see the beginnings of European colonization. In Jamestown in 1907 and in 1920’s the Pilgrams established the Massachusetts Bay Colony.

A huge movement that stretched from the 1550’s all the way until the 1800’s would be The Scientiﬁc Revolution, which was the beginnings of modern science. At this time, developments in the ﬁelds of mathematics, physics, astronomy, biology and chemistry began to transform many views of society and nature.

An English company which formed for the exploitation of trade with East, Southeast Asia, and India which was incorporated by a royal charter at the end of 1600; this multinational company went by the name The East India Trading Company and would later be known as one of the richest trading groups in the world. The company settled down to a trade in cotton, silk, and spices from South India. It extended its reach to the Persian Gulf. In the beginning of the 1620s, the East India Company began to rely on slave labor and transported enslaved people to its areas of inﬂuence in Southeast Asia and India.

Japan became uniﬁed under the rule of Tokugawa Ieyasu, creating the Tokugawa Shogunate which would rule for the next 2 centuries.

17th century China was a rapdly changing region. In the 1600’s, China began to see the inﬂuence of its culture when many Chinese ceramics began to slowly enter the West. The Ming Dynasty was being gradually weakened by factionalism between its civil oﬃcials, the pressure of its rapidly growing

8 population, and a long succession of weak emperors. The Qing Dynasty began to take power in 1636, before the fall of the Ming Dynasty. However, The Ming Dynasty collapsed in 1644, led by a rebel leader named Li Zicheng, and the Qing Dynasty ﬁnally took power.

Signiﬁcant mathematical progress during the Fermat’s lifetime

The 17th century was dubbed the Age of Reason due to some quite remarkable innovations in the mathematical ﬁeld. In the early 17th century, a famous mathematician known as John Napier invented an interesting function which is now known as the Logarithm (log). Later in the 1600’s, Napier and a mathematician named Henry Briggs slowly began to expand further on this logarithm function (log).

Napier was also known for simplifying many calculations, not only for math but also for astronomy and science. Simon Stevin’s Decimal Notation was improved by Napier as well as popularized. Napier made Lattice Multiplication - which was earlier developed by Al-Khwarizmi in Persia and introduced to Europe by Fibonacci - much more convenient with his introduction of ”Napier’s Bones” (a tool which used a set of numbered rods).

Marin Marsenne had great mathematical contributions in this century with his introduction of what is called Marsenne Primes. Marsenne Primes are prime numbers that are one less than a power of 2, such as the following 2, (22 − 1) 3, (23 − 1) Although mathematics was not Marsenne’s prime focus, Marsenne Primes are still a huge tool in discovering more and more primes, even until this day. Yet, Marsenne’s method for discovering primes was not perfect; a mathematician known as Eduoard Lucas had discovered a few ﬂaws. So Lucas eventually came up with a way to check Marsenne Primes and this method is still used to this day.

Another famous mathematician of the 17th century was Rene Descartes, who spent almost 20 years of his life in Germany. Descartes was famous for the development of Analytic Geometry as well as Carte- sian coordinates, which would make it soon possible to plot the orbits of planets. Descartes was also one of the main founders of Calculus and paved a path to a much later ﬁeld of math: Multidimensional Geometry. The use of superscripts for powers was also made popular by Descartes.

Blaise Pascal, mostly known for the Pascal Triangle which is used to determine binomial coefficients, was another huge contributor to the advancement of mathematics in the 1600’s. Along with the Triangle, Pascal was known for the concept of Expected Values as well as Probability Theorem, which he had collaborated with Fermat on. Christiaan Huygens was the first to publish Pascal’s Probability Theorem which was based off of the number of letters shared between both Pascal and Fermat. Huygens also was the first to outline the concept of Mathematical Expectation.

As a successful mathematician and engineer, Girard Desargues played a huge role in laying out the work that led toward the foundation of Projective Geometry -later developed by Jean Victor Poncelet and Gaspard Monge.- Desargues also developed the concept of Point at Inﬁnity, a concept of thinking where parallel lines eventually meet.

Two very famous mathematicians arose at this time, Sir Isaac Newton and Gottfried Leibniz. These two math giants invented infinitesimal calculus in regards to differentiation and integration. A generalized Binomial Theory, the theory of Finite Differences and the use of Infinite Power Series were great contributions by Newton. Leibniz developed a mechanical forerunner to the calculator as well as

9 devising a way to use matrices to solve for Linear Equations. However, a conﬂict arose between the two, Leibniz and Newton were both working on the same ideas, but Leibniz decided to publish the work before Newton. This conﬂict led to Newton’s followers bashing and discrediting Leibniz and accusing him of plagiarism.

The well known symbol of inﬁnity (∞) and the origin of the number line can be connected to a mathematician named John Wallis. In addition to the previously stated, Wallis also extended the standard notation of power where he included negative powers as well as rational numbers. The term Continued Fraction was also introduced by Wallis.

A one Isaac Barrow was the ﬁrst to discover - or at least the ﬁrst statement of - the Fundamental Theorem of Calculus. Barrow also translated many of Euclid’s works into Latin and English all the while being the teacher of the one and only, Father of Calculus, Sir Isaac Newton.

References

1. www.storyofmathematics.com/17th.html

2. www-history.mcs.st-andrews.ac.uk/Biographies/Fermat.html

3. www.math.rutgers.edu/∼ cherlin/History/Papers2000/pellegrino.html

4. www.encyclopedia.com/topic/Pierre de Fermat.aspx

5. www.historylearningsite.co.uk/france-in-the-seventeeth-century/

6. www.britannica.com