Prime Numbers - Base Patterns printable pdf download

Prime Numbers - Base Patterns

Prime Numbers - base patterns...

S.Ferguson

A prime number is exactly divisible

by two different factors:

itself and the number 1

The prime numbers {2, 3, 5, 7, 11, 13... (base ten)} have different written forms

depending on the number base in which they are written. This has prompted many

students to look for patterns for the primes in different bases, but alas! the primes

will not conform, and there is no one formula that will produce every prime num-

ber.

There are patterns to be found, though - in particular, if we look at base six, we find

that the prime numbers from 5 upwards are of form 6n+1 or 6n-1; (but this doesn’t

mean that every number of form 6n±1 is prime).

Here are the first 42 counting numbers arranged in six columns; with the exception

of 2 and 3 the primes, printed in bold italic, fall into columns one and five - i.e.

belonging to numbers of the form 6n±1.

9 10

11 12

15 16 17 18

21 22 23 24

27 28 29 30

33 34 35 36

39 40 41 42

If we rewrite our table, with the numbers now written in base six instead of base

ten, the pattern becomes much more obvious:

5 10

13 14 15 20

23 24 25 30

33 34 35 40

43 44 45 50

53 54 55 100

101

102 103 104 105 110

Other bases of interest are bases four and twelve. Primes in base four are of form

4n±1, and in base twelve 12n±1 and 12n±5.

Prime Numbers - Base Patterns