Confirmation of the non-existence of a projective plane of order 10.

摘要

Mathematicians have always been fascinated with existence problems. Fermat's Last TheoremPsi for example, was posed approximately 300 years before it was finally solved by Andrew Wiles just a few years short of the 21st century, but not before dozens of talented mathematicians (and thousands of not so talented ones) had failed in their attempts to solve it. Another long-standing existence problem, solved just a few years before Fermat's, is the main subject of this thesis: does there exist a projective plane of order 10?;Psi?∃ n > 2, x n + yn = z n, x, y, z, n ∈ Z;We begin laying the necessary groundwork by introducing projective planes and error-correcting codes. Next, a relationship between projective planes and codes is presented as the basis of a method to settle the existence of a projective plane of order 10 (unfortunately in the negative). We also give a historical account of the application of the solution. Finally, a confirmation of the non-existence is implemented and presented.