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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).
As a result, there is a one-to-one correspondence between Mersenne primes and even perfect numbers, so a list of one can be converted into a list of the other. [ 1 ] [ 5 ] [ 6 ] It is currently an open problem whether there are infinitely many Mersenne primes and even perfect numbers.
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.
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.
the number of vertices in a 6-cube, the fourth dodecagonal number, [8] and the seventh centered triangular number. [9] Since it is possible to find sequences of 65 consecutive integers (intervals of length 64) such that each inner member shares a factor with either the first or the last member, 64 is the seventh Erdős–Woods number. [10]
The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) 2 = x 2 ...
Perfect square dissection, a dissection of a geometric square into smaller squares, all of different sizes Perfect square trinomials , a method of factoring polynomials Topics referred to by the same term