Fermat's Last Theorem: Its Proofs
Fermat's Last Theorem (FLT) has remained most famous unsolved mathematical
problem. This theorem has also attracted the attention of Dr.S.K.Kapoor. He has
published his results about the truth of this property of numbers. He has worked
out the theorem with different approaches. Essentially all these approaches are
geometric in nature but still these are different in characteristics and as such
these may be taken as different proofs of the theorem. His approaches and
conclusions are not only having intrinsic academic values but also these have
historic angle as much as that these accept inspiration as well as working
operations from a distant source i.e. Vedic literature. These results are also
published in point of time than the proof now accepted in the academic circles.
Three of these proofs of FLT were published in "Modern Science & Vedic Science"
(Volume 3 No.1, 1989) with the caption "Vedic Mathematical Concepts and Their
Applications to Unsolved Mathematical Problems: Three Proofs of Fermat's Last
Theorem". Chapter-10 of the book "Vedic Geometry" (1994) as well takes up some
of the aspects of this property of numbers under the caption "Chapter 10:
Conclusions and their Applications to the Solution of Fermat's Last Theorem".
Then followed the book "Fermat's Last Theorem and Higher Spaces Reality Course"
(1996) in which in addition to the outline of different formats of the
approaches to the proof and in addition to the general proof of FLT, the result
of the theorem has been generalised as Generalised FLT for whole range of
hypercubes. The concepts of power sets and the application of different place
value systems to test the truth of any given triple of whole numbers have also
been introduced which may prove to be very handy for tests of the truth of this
property known as FLT. These proofs are on different formats which are worked
out as:
1. On Domain Format
2. On Simplex Format
3. On Values Square Format
4. On the Format of Domain As Dimension Of Another Domain
5. On the Format of Hypercubes
Dr. Kapoor has generalised the property by extending it to the geometric
property of hypercubes of every order. Fermat's Last Theorem speaks out the
property of linear dimensional order only. The cause of the restriction being
the linear dimensional order so we get the restriction of powers being three and
higher. Dr. Kapoor's results generalized the theorem as a geometric property of
hypercubes by a shift from linear dimensional order to a spatial dimensional
order. The generalized statement comes to be as that "no hypercube can be
duplicated" because of the interlocking of (n-2) space with n-space as dimension
and domain respectively.
For practical testing of triplets of whole numbers Dr. Kapoor has introduced the
concept of Power Sets and the same
have been tabulated as Appendix of the book.