Pierre De Fermat
Pierre de Fermat Pierre de Fermat was born
in the year 1601 in Beaumont-de-Lomages,
France. Mr. Fermat's education
began in 1631. He was home schooled. Mr. Fermat
was a single man through his
life. Pierre de Fermat, like many mathematicians of
the early 17th century,
found solutions to the four major problems that created
a form of math called
calculus. Before Sir Isaac Newton was even born, Fermat
found a method for
finding the tangent to a curve. He tried different ways in
math to improve
the system. This was his occupation. Mr. Fermat was a good
scholar, and
amused himself by restoring the work of Apollonius on plane loci.
Mr.
Fermat published only a few papers in his lifetime and gave no
systematic
exposition of his methods. He had a habit of scribbling notes in
the margins of
books or in letters rather than publishing them. He was modest
because he
thought if he published his theorems the people would not believe
them. He did
not seem to have the intention to publish his papers. It is
probable that he
revised his notes as the occasion required. His published
works represent the
final form of his research, and therefore cannot be dated
earlier than 1660. Mr.
Pierre de Fermat discovered many things in his
lifetime. Some things that he did
include: -If p is a prime and a is a prime
to p then ap-1-1 is divisible by p,
that is, ap-1-1=0 (mod p). The proof of
this, first given by Euler, was known
quite well. A more general theorem is
that a0-(n)-1=0 (mod n), where a is prime
to n and p(n) is the number of
integers less than n and prime to it. -An odd
prime number can be expressed
as the difference of two square integers in only
one way. Fermat's proof is
as follows. Let n be prime, and suppose it is equal
to x2 -y2 that is, to
(x+y)(x-y). Now, by hypothesis, the only basic, integral
factors of n and n
and unity, hence x+y=n and x-y=1. Solving these equations we
get x=1 /2 (n+1)
and y=1 /2(n-1). -He gave a proof of the statement made by
Diophantus
that the sum of the squares of two numbers cannot be the form of
4n-1. He
added a corollary which I take to mean that it is impossible that the
product
of a square and a prime form 4n-1[even if multiplied by a number that
is
prime to the latter], can be either a square or the sum of two squares.
For
example, 44 is a multiple of 11(which is of the form 4 x 3 - 1) by 4,
therefore
it cannot be expressed as the sum of two squares. He also stated
that a number
of the form a2 +b2, where a is prime b, cannot be divided by a
prime of the form
4n-1. -Every prime of the form 4n+1 is accurate as the
sum of two squares. This
problem was first solved by Euler, who showed that a
number of the form 2(4n+1)
can be always showen as the sum of two squares, of
course it was Mr. Pierre de
Fermat. -If a, b, c, are integers, a2 + b2=
c2, then ab cannot be a square.
Lagrange solved this. - The determination
of a number x such that x2n+1 may be
squared, where n is a given integer
which is not squared. Lagrange gave a
solution of this also. -There is only
one integral solution of the equation x2
+4=y3. The required solutions are
clearly for the first equation x=5, and for
the second equation x=2and x=11.
This question was issued as a challenge to the
English mathematicians
Wallis and Digby. -No basic values of x, y, z can be
found to satisfy the
equation xn+yn=zn; if n is an integer greater than 2. This
thesis has
achieved extraordinary celebrity from the fact that no general
demonstration
of it has been given, but there is no reason to doubt that this
true. -Fermat
also discovered the general theorem that was on the guess that a
number can
be found into the product of powers of primes in only one way. These
were
some interesting things that Mr. Fermat did in his life. During
Mr.
Fermat's life many things happened as world events. First Ludolph Van
Ceulen
died, there is a site dedicated to this long-ignored mathematician,
who spent
his entire life, approximating Pi to 35 places. Then Blaise Pascal
lived his
entire life, born in 1623 and died in 1662. Next Sir Isaac Newton
was born in
the year 1642. Then Marin Mersenne, French philosopher,
mathematician, Jesuit
theologian, and scientist died in the year 1648.
Finally Mr. Pierre de Fermat
died in the year 1665. Some of the most striking
results were discovered after
his death on loose sheets of paper or written
in the margins of works which he
had read and annotated. These annotations
are unsupported by proof. Scholars
cannot say that Mr. Fermat's theorems are
positive until proof is found.