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In mathematics, Hall's conjecture is an open question on the differences between perfect squares and perfect cubes.It asserts that a perfect square y 2 and a perfect cube x 3 that are not equal must lie a substantial distance apart.
Square number 16 as sum of gnomons. In mathematics, a square number or perfect square is an integer that is the square of an integer; [1] in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 3 2 and can be written as 3 × 3.
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).
Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4]
Magic square. Frénicle standard form; Prime reciprocal magic square; Trimagic square; Multimagic square; Panmagic square; Satanic square; Most-perfect magic square; Geometric magic square; Conway's Lux method for magic squares; Magic cube. Perfect magic cube; Semiperfect magic cube; Bimagic cube; Trimagic cube; Multimagic cube; Magic hypercube ...
Perfect: All 3m planar arrays must be pandiagonal magic squares. In addition, all pantriagonals must sum correctly. These two conditions combine to provide a total of 9m pandiagonal magic squares. The smallest normal perfect magic cube is order 8; see Perfect magic cube. Nasik; A. H. Frost (1866) referred to all but the simple magic cube as Nasik!
Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a 2 b 3, where a and b are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful.
Cubing the cube is the analogue in three dimensions of squaring the square: that is, given a cube C, the problem of dividing it into finitely many smaller cubes, no two congruent. Unlike the case of squaring the square, a hard yet solvable problem, there is no perfect cubed cube and, more generally, no dissection of a rectangular cuboid C into ...