Early Work
What major did Fermat study while he was in school?
Fermat received most of his education in a local Franciscan school. He studied law during this time.
What did these curves direct Fermat to do during the middle 1630s?
They directed Fermat to find an algorithm that was equivalent to differentiation.
What new results did Fermat discover in the higher arithmetic?
Many properties Fermat discovered consisted of prime numbers (those positive integers that have no factors other than 1 and themselves).
What law did Fermat's opinion differ from in comparison to the Cartesian views?
Fermat differed with Cartesian views concerning the law of refraction (the sines of the angles of incidence and refraction of light passing through media of different densities are in a constant ratio).
Is it true that Fermat generalized the Archimedean spiral?
True, the formula is r = aθ.
What beliefs did Fermat follow?
Fermat followed the custom of his day in creating conjectural “restorations” of lost works of antiquity. Foreign languages, classical literature, and ancient science and mathematics mainly influenced his developed interests.
What did Fermat's study of curves prompt him to do?
Fermat’s study of curves and equations prompted him to create the equation for the ordinary parabola ay = x2, and that for the rectangular hyperbola xy = a2, to form an - 1y = xn.
What is the form of every prime number according to Fermat?
4n + 1
Why did Fermat note that the sine law appear to be in conflict with the view espoused by Aristotelians?
Fermat argued that nature always chooses the shortest path, which does not favor the opinion of the Aristotelians.
Is it true that the formula Fermat discovered through a summation process is still used today?
True, the formula found is equivalent to the formula currently used for the same purpose in integral calculus.
What did Fermat accomplish by 1629?
Fermat had begun a reconstruction of the long-lost Plane Loci of Apollonius, which was the Greek geometer of the 3rd century BCE.
How did Fermat's procedures influence his findings for equations of tangents?
This procedure enabled Fermat to find equations of tangents to curves and to locate the maximum, minimum, and inflection points of polynomial curves.
What is the result of Fermat's theory? What does it express?
Every prime of the form 4n + 1 is uniquely expressible as the sum of two squares.
What methods did Fermat apply to making the assumption that light travels less rapidly in the denser medium?
It is known that Fermat noticed that the differentiation of xn, leading to nan - 1, is the inverse of integrating xn.
False, it is not known whether or not Fermat knew this information.
What did Fermat discover through the study of Loci?
Fermat discovered that sets of points with certain characteristics could be facilitated by applying algebra to geometry through a coordinate system.
What are the curves determined by this equation known as?
The curves determined by his equation are known as the parabolas or hyperbolas of Fermat, referred to as n is either positive or negative.
What does Fermat’s lesser theorem assert?
Fermat’s lesser theorem asserts that if p is a prime number and if a is any positive integer, then ap - a is divisible by p.
What relation did Fermat make with that the of the law of refraction?
Fermat showed that the law of refraction is consonant with his “principle of least time.”
Is it true that Fermat’s lesser theorem asserts that if p is a prime number and if a is any negative integer, then ap - a is divisible by p?
False, a has to be a positive integer.
How did Descartes' and Fermat's findings help push forth their discoveries?
Descartes had evaluated the same basic principle of analytic geometry: equations in two variable quantities define plane curves. Because Fermat’s Introduction to Loci was published in 1679, the exploitation of their discovery, initiated in Descartes’s Géométrie of 1637, has since been known as Cartesian geometry.
What kind of problems did Fermat handle through his transformations?
Fermat handled problems involving more general algebraic curves.
When Fermat announced his belief that numbers of form 22n + 1, known since as “numbers of Fermat,” are necessarily prime, what happened a century later?
Euler showed that 225 + 1 has 641 as a factor.
Who was Fermat's argument concerning the speed of light in agreement with? What theory did this person present?
Fermat's argument concerning the speed of light was later found to be in agreement with the wave theory of the 17th-century Dutch scientist Christiaan Huygens, and in 1849 it was verified experimentally by A.-H.-L. Fizeau.
Is it true that Descartes had sought to justify the sine law through a premise that light travels more rapidly in the denser of the two media involved in the refraction?
True