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This work is dedicated to the properties of the 3 × 3 magic squares of cubes modulo a prime number. Its central concept is the number of distinct entries of these squares and the properties associated with this number. We call this number the degree of a magic square. The necessary conditions for the magic square of cubes with degrees 3, 5, 7, and 9 are examined. It was established that there are infinitely many primes for which magic squares of cubes with degrees 3, 5, 7, and 9 exist. I apply n-tuples of consecutive cubic residues to prove that there are infinitely many Magic Squares of Cubes with degree 9. Furthermore I use Brauer’s theorem, that guarantees the existence of a sequence of consecutive integers of any length, to construct Magic Squares of Cubes whose entries are all cubic residues. Such analytic tools as Modular Arithmetic, Legendre symbol, Fermat’s Little Theorem, notions of quadratic and cubic residues were employed in the process of research.
No advanced mathematical knowledge to construct these three-dimensional mind bogglers; including pandiagonal and perfect cubes ? many entirely new constructions, too. 111 figures.
Excerpt from Magic Squares and Cubes Magic squares are a visible instance of the intrinsic harmony of the laws of number, and we are thrilled with joy at beholding this evidence which re ects the glorious symmetry of the cosmic order. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works."