## 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.