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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.
The sum of Euler's totient function φ(x) over the first twenty integers is 128. [4] 128 can be expressed by a combination of its digits with mathematical operators, thus 128 = 2 8 − 1, making it a Friedman number in base 10. [5] A hepteract has 128 vertices. 128 is the only 3-digit number that is a 7th power (2 7).
Fractions: A representation of a non-integer as a ratio of two integers. These include improper fractions as well as mixed numbers . Continued fraction : An expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of ...
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 ...
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.
Likewise, the second largest black square has area 1 / 16 , and the third largest black square has area 1 / 64 . The area taken up by all of the black squares together is therefore 1 / 4 + 1 / 16 + 1 / 64 + ⋯ , and this is also the area taken up by the gray squares and the white squares.
All square triangular numbers have the form , where is a convergent to the continued fraction expansion of , the square root of 2. [ 4 ] A. V. Sylwester gave a short proof that there are infinitely many square triangular numbers: If the n {\displaystyle n} th triangular number n ( n + 1 ) 2 {\displaystyle {\tfrac {n(n+1)}{2}}} is square, then ...
The sum of the reciprocals of the divisors of a perfect number is 2. When eight points are distributed on the surface of a sphere with the aim of maximizing the distance between them in some sense, the resulting shape corresponds to a square antiprism. Specific methods of distributing the points include, for example, minimizing the sum of all ...