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With even cubes, there is considerable restriction, for only 00, o 2, e 4, o 6 and e 8 can be the last two digits of a perfect cube (where o stands for any odd digit and e for any even digit). Some cube numbers are also square numbers; for example, 64 is a square number (8 × 8) and a cube number (4 × 4 × 4).
Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2n + 1) 2 = 4n(n + 1) + 1, and n(n + 1) is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8. Every odd perfect square is a centered octagonal number. The difference between any two odd perfect squares is a multiple of 8.
Hall's conjecture. In mathematics, Hall's conjecture is an open question on the differences between perfect squares and perfect cubes. It asserts that a perfect square y2 and a perfect cube x3 that are not equal must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theory of integer points ...
The primary square is obtained by rotating the root square counter-clockwise by 90 degrees, and replacing the numbers. The resulting square is an associative magic square, in which every pair of numbers symmetrically opposite to the center sum up to the same value, 26. For e.g., 16+10, 3+23, 6+20, etc.
Magic hypercube. In mathematics, a magic hypercube is the k -dimensional generalization of magic squares and magic cubes, that is, an n × n × n × ... × n array of integers such that the sums of the numbers on each pillar (along any axis) as well as on the main space diagonals are all the same. The common sum is called the magic constant of ...
Perfect magic cube. In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars, and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant. [1][2][3] Perfect magic cubes of order one are trivial; cubes of orders two to four can be proven not to exist, [4] and cubes ...
The difference of two squares is used to find the linear factors of the sum of two squares, using complex number coefficients. For example, the complex roots of can be found using difference of two squares: (since ) Therefore, the linear factors are and . Since the two factors found by this method are complex conjugates, we can use this in ...
The 6 oblique squares are also simple magic. The smallest normal diagonal magic cube is order 5. These squares were referred to as 'Perfect' by Gardner and others. At the same time he referred to Langman’s 1962 pandiagonal cube also as 'Perfect'. Christian Boyer and Walter Trump now consider this and the next two classes to be Perfect.