CHONKANYANUKOON SCHOOL
Semester 2
Additional Mathematics (M32101)
English Program 4/1
SUBMITTED BY:GROUP 4
1.PHIPHATNAREE
T.- NO.06 EP4/1
2.WARATCHAYA
T.- NO.08 EP4/1
3.SUPARPICH
S.- NO.32 EP4/1
4.SUPISSARA
S.- NO.33 EP4/1
SUBMITTED TO:
TEACHER :WENDEL GLENN JUMALON
(MATHEMATICS TEACHER)
I. Introduction
The famous "Last Theorem" for which Fermat is best know by students is not used
nearly so often as the one which is remembered as his "little" theorem. The little theorem is
often used in number theory in the testing of large primes and simply states that: if p is a
prime which does not divide a, then ap-1=1 (mod p) . In more simple language this says that if
p is a prime that is not a factor of a, then when a is multiplied together p-1 times, and the
result divided by p, we get a remainder of one. For example, if we use a=7 and p=3, the rule
says that 72 divided by 3 will have a remainder of one. In fact 49/3 does have a remainder of
one. The theorem was first stated by Fermat in a letter in 1640 without a proof. Euler gave the
first published proof in 1736. Here is a link to a proof of the theorem.
The theorem is a one direction theorem, what mathematicians call "necessary, but not
sufficient". What that means is that although it is true for all primes, it is not true JUST for
primes, and will sometimes be true for other numbers as well. For example 390 =1 (mod 91),
but 91 is not prime.
We can test 390 using Fermat's Little Theorem without ever finding out what the
actual value of 390 is, by using the patterns of remainders for powers of 3 divided by 91...
3^1 = 3 remainder on division is 3
3^2 = 9
........................
9
3^3=27
........................
27
3^4=81
........................
81
3^5=243 ........................
61
3^6= 729 .......................
1
Since 36 = 1 (mod 91) then any power of 36 will also be =1 (mod 91), and 390= (36)15.
Numbers which meet the conditions of Fermat's Little Theorem but are not prime are
called pseudoprimes, or probable primes relative base n. Although 91 is a
pseudoprime base three, it does not work for other bases. For mathematicians testing large
numbers, it is much easier to test against several bases using ap-1 then to try to factor huge
numbers. Since there are relatively few pseudoprimes, and even less that are pseudoprimes to
more than one base, this often works. There are, however, some numbers that are
pseudoprimes to every base to which they are relatively prime. These most difficult of
pseudoprimes are called Carmichael numbers. Since there is little written about Carmichael
on the net, I include a rather nice capsule history of his life and work presented by Julio
Cabillon to the Math History Newsgroup a few years ago.
II. Background
History of Fermat’s little theorem are discovered by Pierre de Fermat found the
theorem around 1636. It appeared in one of his letters, dated October 18, 1640 to his
confidant Frenicle as the following: p divides whenever p is prime and a is coprime to p.
Chinese mathematicians independently made the related hypothesis (sometimes called
the Chinese Hypothesis) that p is a prime if and only if . It is true that if p is prime, then (this
is a special case of Fermat's little theorem). However, the converse (if then p is prime), and
therefore the hypothesis as a whole, is false (e.g. 341=11×31 is a pseudoprime, see below).
It is widely stated that the Chinese hypothesis was developed about 2000 years before
Fermat's work in the 1600's. Despite the fact that the hypothesis is partially incorrect, it is
noteworthy that it may have been known to ancient mathematicians. Some, however, claim
that the widely propagated belief that the hypothesis was around so early sprouted from a
misunderstanding, and that it was actually developed in 1872. For more on this, see
(Ribenboim, 1995).
Pierre de Fermat (French pronunciation: 17 August 1601 or 1607/8[1] – 12 January
1665) was a French lawyer at the Parlement of Toulouse, France, and an amateur
mathematician who is given credit for early developments that led to infinitesimal calculus.
In particular, he is recognized for his discovery of an original method of finding the greatest
and the smallest ordinates of curved lines, which is analogous to that of the then unknown
differential calculus, as well as his research into the theory of numbers. He made notable
contributions to analytic geometry, probability, and optics. He is best known for Fermat's
Last Theorem, which he described in a note at the margin of a copy of Diophantus'
Arithmetica.
A.Life and work
Fermat was born in Beaumont-de-Lomagne, Tarn-et-Garonne, France; the late
15th century mansion where Fermat was born is now a museum. He was of Basque
origin. Fermat's father was a wealthy leather merchant and second consul of
Beaumont-de-Lomagne. Pierre had a brother and two sisters and was almost certainly
brought up in the town of his birth. There is little evidence concerning his school
education, but it may have been at the local Franciscan monastery.
He attended the University of Toulouse before moving to Bordeaux in the
second half of the 1620s. In Bordeaux he began his first serious mathematical
researches and in 1629 he gave a copy of his restoration of Apollonius's De Locis
Planis to one of the mathematicians there. Certainly in Bordeaux he was in contact
with Beaugrand and during this time he produced important work on maxima and
minima which he gave to Étienne d'Espagnet who clearly shared mathematical
interests with Fermat. There he became much influenced by the work of Franciscus
Vieta.
From Bordeaux, Fermat went to Orléans where he studied law at the
University. He received a degree in civil law before, in 1631, receiving the title of
councillor at the High Court of Judicature in Toulouse, which he held for the rest of
his life. Due to the office he now held he became entitled to change his name from
Pierre Fermat to Pierre de Fermat. Fluent in Latin, Basque[citation needed], classical
Greek, Italian, and Spanish, Fermat was praised for his written verse in several
languages, and his advice was eagerly sought regarding the emendation of Greek
texts.
He communicated most of his work in letters to friends, often with little or no
proof of his theorems. This allowed him to preserve his status as an "amateur" while
gaining the recognition he desired. This naturally led to priority disputes with fellow
contemporaries such as Descartes and Wallis. He developed a close relationship with
Blaise Pascal.
Anders Hald writes that, "The basis of Fermat's mathematics was the classical
Greek treatises combined with Vieta's new algebraic methods.”
a).Work
Fermat's pioneering work in analytic geometry was circulated in manuscript
form in 1636, predating the publication of Descartes' famous La géométrie. This
manuscript was published posthumously in 1679 in "Varia opera mathematica", as Ad
Locos Planos et Solidos Isagoge, ("Introduction to Plane and Solid Loci").
In Methodus ad disquirendam maximam et minima and in De tangentibus
linearum curvarum, Fermat developed a method for determining maxima, minima,
and tangents to various curves that was equivalent to differentiation. In these works,
Fermat obtained a technique for finding the centers of gravity of various plane and
solid figures, which led to his further work in quadrature.
Fermat was the first person known to have evaluated the integral of general
power functions. Using an ingenious trick, he was able to reduce this evaluation to the
sum of geometric series. The resulting formula was helpful to Newton, and then
Leibniz, when they independently developed the fundamental theorem of calculus.
In number theory, Fermat studied Pell's equation, perfect numbers, amicable
numbers and what would later become Fermat numbers. It was while researching
perfect numbers that he discovered the little theorem. He invented a factorization
method - Fermat's factorization method - as well as the proof technique of infinite
descent, which he used to prove Fermat's Last Theorem for the case n = 4. Fermat
developed the two-square theorem, and the polygonal number theorem, which states
that each number is a sum of three triangular numbers, four square numbers, five
pentagonal numbers, and so on.
Although Fermat claimed to have proved all his arithmetic theorems, few
records of his proofs have survived. Many mathematicians, including Gauss, doubted
several of his claims, especially given the difficulty of some of the problems and the
limited mathematical tools available to Fermat. His famous Last Theorem was first
discovered by his son in the margin on his father's copy of an edition of Diophantus,
and included the statement that the margin was too small to include the proof. He had
not bothered to inform even Marin Mersenne of it. It was not proved until 1994, using
techniques unavailable to Fermat.
Although he carefully studied, and drew inspiration from Diophantus, Fermat
began a different tradition. Diophantus was content to find a single solution to his
equations, even if it were an undesired fractional one. Fermat was interested only in
integer solutions to his Diophantine equations, and he looked for all possible general
solutions. He often proved that certain equations had no solution, which usually
baffled his contemporaries.
Through his correspondence with Pascal in 1654, Fermat and Pascal helped
lay the fundamental groundwork for the theory of probability. From this brief but
productive collaboration on the problem of points, they are now regarded as joint
founders of probability theory. Fermat is credited with carrying out the first ever
rigorous probability calculation. In it, he was asked by a professional gambler why if
he bet on rolling at least one six in four throws of a die he won in the long term,
whereas betting on throwing at least one double-six in 24 throws of two dice resulted
in him losing. Fermat subsequently proved why this was the case mathematically.
Fermat's principle of least time (which he used to derive Snell's law in 1657)
was the first variational principle enunciated in physics since Hero of Alexandria
described a principle of least distance in the first century CE. In this way, Fermat is
recognized as a key figure in the historical development of the fundamental principle
of least action in physics. The term Fermat functional was named in recognition of
this role.
b). Death
He died at Castres, Tarn. The oldest, and most prestigious, high school in
Toulouse is named after him: the Lycée Pierre de Fermat. French sculptor Théophile
Barrau made a marble statue named Hommage à Pierre Fermat as tribute to Fermat,
now at the Capitole of Toulouse.
B. Assessment of his work
Together with René Descartes, Fermat was one of the two leading
mathematicians of the first half of the 17th century. Independently of Descartes, he
discovered the fundamental principles of analytic geometry. With Blaise Pascal, he
was a founder of the theory of probability.
Regarding Fermat's work in analysis, Isaac Newton wrote that his own early
ideas about calculus came directly from "Fermat's way of drawing tangents."
Of Fermat's number theoretic work, the great 20th century mathematician
André Weil wrote that "... what we possess of his methods for dealing with curves of
genus 1 is remarkably coherent; it is still the foundation for the modern theory of such
curves. It naturally falls into two parts; the first one ... may conveniently be termed a
method of ascent, in contrast with the descent which is rightly regarded as Fermat's
own." Regarding Fermat's use of ascent, Weil continued "The novelty consisted in the
vastly extended use which Fermat made of it, giving him at least a partial equivalent
of what we would obtain by the systematic use of the group theoretical properties of
the rational points on a standard cubic." With his gift for number relations and his
ability to find proofs for many of his theorems, Fermat essentially created the modern
theory of numbers.
III. The report
Fermat's little theorem states that if P is a prime number, then for any integer n ≥1.
Theorem:
(FLP): Let P is a prime number then,
nP ≡ n(mod p)
Corollary:
Let P is a prime number then,
nP-1 ≡ 1(mod p)
For any integer n ≥1 with (n, p) = 1
A. Example
Example 1:
(FLP): Let P is a prime number then,
nP ≡ n(mod p)
n = 5, p = 3
53
≡ 5(mod 3)
125 ≡ 5(mod 3)
125-5 is divisible by 3
3/125-5
Corollary 1:
Let P is a prime number then,
nP-1 ≡ 1(mod p)
For any integer n ≥1 with (n, p) = 1
53-1
≡ 5(mod 3)
52
≡ 5(mod 3)
25
≡ 5(mod 3)
Example 2:
(FLP) : Let P is a prime number then,
nP ≡ n(mod p)
n = 7, p = 2
72 ≡ 7(mod 2)
49 ≡ 7(mod 2)
49-7 is divisible by 2
2/49-7
Corollary 2:
Let P is a prime number then,
nP-1 ≡ 1(mod p)
For any integer n ≥1 with (n, p) = 1
72-1
≡ 7(mod 2)
71
≡ 7(mod 2)
7
≡ 7(mod 2)
B.Activity
Activity 1:
(FLP) : Let P is a prime number then,
nP ≡ n(mod p)
n = 9, p = 3
93
≡ 9(mod 3)
729 ≡ 9(mod 3)
729-9 is divisible by 3
3/729-9
Corollary :
Let P is a prime number then,
nP-1 ≡ 1(mod p)
For any integer n ≥1 with (n.p) = 1
93-1
≡ 1(mod 3)
92
≡ 1(mod 3)
81
≡ 1(mod 3)
Activity 2:
(FLP) : Let P is a prime number then,
nP ≡ n(mod p)
n = 15, p = 3
153
≡ 15(mod 3)
3375
≡ 15(mod 3)
3375-15 is divisible by 3
3/3375-15
Corollary :
Let P is a prime number then,
nP-1 ≡ 1(mod p)
For any integer n ≥1 with (n.p) = 1
153-1
≡ 15(mod 3)
152
≡ 15(mod 3)
225
≡ 15(mod 3)
Activity 3:
(FLP) : Let P is a prime number then,
nP ≡ n(mod p)
n = 20, p = 2
202
≡ 20(mod 2)
400
≡ 20(mod 2)
400-20 is divisible by 2
2/400-20
Corollary 3 :
Let P is a prime number then,
nP-1 ≡ 1(mod p)
For any integer n ≥1 with (n.p) = 1
202-1
≡ 20(mod 2)
201
≡ 20(mod 2)
20
≡ 20(mod 2)
IV. Summary
We observed that by using this theorem, we can adapt it into our daily life
and can use this theorem in worldwide and in many occupations.
For example, by using this theorem that is “Fermat's Little Theorem”. Not only
that, but we also observed that by using this theorem, we can find the missing
number in many ways and all the numbers have connection to each other.
VI. References
http://www.pballew.net/FermLit.html
http://en.wikipedia.org/wiki/Pierre_de_Fermat
http://www.experiencefestival.com/a/Fermats_little_theorem_-_History/id/5044755