Sierpiński's constant

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Sierpiński's constant is a mathematical constant usually denoted as K. One way of defining it is as the following limit:

where r2(k) is a number of representations of k as a sum of the form a2 + b2 for integer a and b.

It can be given in closed form as:

where is Gauss's constant and is the Euler-Mascheroni constant.

Another way to define/understand Sierpiński's constant is,

Graph of the given equation where the straight line represents Sierpiński's constant.

Let r(n)[1] denote the number of representations of  by  squares, then the Summatory Function[2] of has the Asymptotic[3] expansion


where  is the Sierpinski constant. The above plot shows


with the value of  indicated as the solid horizontal line.

See also[edit]

External links[edit]

  • [1]
  • - Sierpiński's constant up to 2000th decimal digit.
  • Weisstein, Eric W. "Sierpinski Constant". MathWorld.
  • OEIS sequence A062089 (Decimal expansion of Sierpiński's constant)


  1. ^ "r(n)". Retrieved 2021-11-30.
  2. ^ "Summatory Function". Retrieved 2021-11-30.
  3. ^ "Asymptotic". Retrieved 2021-11-30.