MATHS BITE: Sierpinski Number

A Sierpinski number is an odd natural number k such that {\displaystyle k\times 2^{n}+1} is not prime for all natural numbers n. In 1960, Sierpinski proved that there are infinitely many odd integers k with this property, but failed to give an example. Numbers in such a set with odd k and k < 2^n are called Proth numbers.

78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909,….

-List of some known Sierpinski Numbers

Sierpinski Problem

The Sierpinski problem asks what the smallest Sierpinski number is. In 1967, Sierpinski and Selfridge conjectured that 78557 is the smallest Sierpinski number. To show this is true, we need to show that all the odd numbers smaller than 78557 are not Sierpinski numbers, i.e. for every odd k below 78557 there is a positive interger n such that {\displaystyle k\times 2^{n}+1} is prime. There are only five numbers which have not been eliminated:

k = 21181, 22699, 24737, 55459, and 67607

Numberphile Video

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The Sierpinski Triangle

The Sierpinski Triangle, or Sierpinski Sieve, is a fractal described by Polish Mathematician Sierpinski in 1915, although it appeared in Italian art from the 13th century. It has an overall shape of an equilateral triangle, and is subdivided recursively into smaller equilateral triangles.

Sierpinski sieve from rule 90
Source: Wolfram Mathworld

Constructing a Sierpinski Triangle


Start with an equilateral triangle.


Connect the midpoints of each side, hence dividing it into 4 smaller congruent equilateral triangles.



Now cut out the triangle in the centre.

Step One


Repeat steps 2 and 3 with each of the remaining smaller triangles.

[Sierpinski Triangle]



If we let N_n be the number of black triangles after iteration nL_n be the length of a side of a triangle, and A_n be the fractional area which is black after the nth iteration, then:

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