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Fermat Number, a class of numbers, is an integer of the form .
For example: Putting in we get , , , etc.
Fermat observed that all the integers were prime numbers and announced that is a prime for each natural value of .
In writing to Prof. Mersenne, Fermat confidently announced:
I have found that numbers of the form are always prime numbers and have long since signified to analysts the truth of this theorem.
However, he also accepted that he was unable to prove it theoretically. Euler in 1732 negated Fermat’s fact and told that are primes but is not a prime since it is divisible by 641.
Euler also stated that all Fermat numbers are not necessarily primes and the Fermat number which is a prime, might be called a Fermat Prime. Euler used division to prove the fact that is not a prime. The elementary proof of Euler’s negation is due to G. Bennett.
|The Fermat number is divisible by i.e., .|
Factorising in such a way that
Subtracting from 641, we get .
Now again, equation (1) could be written as
Mathematics is on its progression and well developed now but it is yet not confirmed that whether there are infinitely many Fermat primes or, for that matter, whether there is at least one Fermat prime beyond . The best guess is that all Fermat numbers are composite (non-prime).
A useful property of Fermat numbers is that they are relatively prime to each other; i.e., for Fermat numbers , .
Following two theorems are very useful in determining the primality of Fermat numbers:
|For , the Fermat number is prime|
Euler- Lucas Theorem
|Any prime divisor of , where , is of form .|
Fermat numbers () with are prime; with have completely been factored; with have two or more prime factors known; with have only one prime factor known; with have no factors known but proved composites. has not yet been proved either prime or composite.
- Video: Documentary on Proof of Fermat’s Last Theorem (wpgaurav.wordpress.com)
Real projective n-space
Real projective space, is defined to be the space of all lines through the origin in . Each such kind is determined by a non-zero vector in , unique up to scalar multiplication, and is topologized as the quotient space of under the equivalence relation for scalars . We can restrict to vectors of length 1, so is also the quotient spaces , the sphere with antipodal points identified. This is equivalent to saying that is the quotient space of a hemisphere with antipodal points of identified. Since with antipodal points identified is just , we see that is obtained from by attaching an -cell, with the quotient projection as the attaching map. It follows by induction on that has a cell complex structure with one cell in each dimension . Since is obtained from by attaching an -cell, the infinite union becomes a Cell Complex with one cell in each dimension. We can view as the space of lines through the origin in .
Complex Projective n-space
Complex projective n-space is space of complex lines through the origin in , that is, 1-dimensional vector subspaces of . As in the case of Real projective n-space, each line is determined by a non-zero vector in, unique up to scalar multiplication, and is topologized as the quotient space of under the equivalence relation for .
Equivalently, this is the quotient of the unit Sphere with for .
It is also possible to obtain as a quotient space of disk under the identifications for , in the following way:
The vectors in with last coordinate real and nonnegative are precisely the vectors of the form with . Such vectors form the of the function . This is a disk bounded by the sphere consisting of vectors with . Each vector in is equivalent under the identifications to a vector in , and the latter vector is unique if its last coordinate is non-zero. If the last coordinate is zero, we have just the identifications for .
From the description of as the quotient of under the identifications for , it follows that is obtained from by attaching a cell via the quotient map . So by induction on , we obtain a cell structure
with cells only in even dimensions. Similarly, has a cell structure with one cell in each even dimension.
How to Draw a Cell Complex (or CW Complex)
Let we try to construct a space by following procedure:
- Start with a discrete set , whose points are regarded as -cells.
- Inductively, form the -skeleton from by attaching -cells via maps . This means that is the quotient space of the disjoint union of with a collection of n-disks under the identifications for . Thus as a set
where each is an open disk.
- One can either stop this inductive process at a finite stage, setting for some , one can continue indefinitely, setting .
A space constructed in this way is called a CELL COMPLEX or CW COMPLEX.
A series of positive terms is convergent if from and after some fixed term , where r is a fixed number. The series is divergent if from and after some fixed term.
D’ Alembert’s Test is also known as ratio test of convergency of a series.
Definitions for Generally Interested Readers
(Definition 1) An infinite series i.e. is said to be convergent if , the sum of its first terms, tends to a finite limit as n tends to infinity.
We call the sum of the series, and write .
Thus an infinite series converges to a sum S, if for any given positive number , however small, there exists a positive integer such that
for all .
If as , the series is said to be divergent.
Thus, is said to be divergent if for every given positive number , however large, there exists a positive integer such that for all .
If does not tends to a finite limit, or to plus or minus infinity, the series is called Oscillatory
Let a series be . We assume that the above inequalities are true.
- From the first part of the statement:
, ……… where r <1.
Since r<1, therefore as
therefore =k say, where k is a fixed number.
Therefore is convergent.
- Since, then, , …….
Therefore and so on.
Therefore > . By taking n sufficiently large, we see that can be made greater than any fixed quantity.
Hence the series is divergent.
- When , the test fails.
Another form of the test–
The series of positive terms is convergent if >1 and divergent if <1.
One should use this form of the test in the practical applications.
Verify whether the infinite series is convergent or divergent.
We have and
Hence, when 1/x >1 , i.e., x <1, the series is convergent and when x >1 the series is divergent.
Take Now , a non-zero finite quantity.
But is convergent.
Hence, is also Convergent.