## MATHS BITE: Sierpinski Number

A Sierpinski number is an odd natural number k such that  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  is prime. There are only five numbers which have not been eliminated:

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

### Numberphile Video

Find out more here. M x

## 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.

### Constructing a Sierpinski Triangle

STEP 1:

STEP 2:

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

STEP 3:

Now cut out the triangle in the centre.

STEP 4:

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

### Properties

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

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