Prime Numbers - Base Patterns (Page 3 of 3) in pdf

Prime Numbers - Base Patterns Page 3

ASE

WELVE AND THE

RIME

UMBERS

(by Don Hammond, in Dozenal Journal no. 5)

The dozenal base quickly and easily reveals a fundamental property of prime numbers.

For natural numbers in base twelve greater than 3:

• All numbers terminating with even digits are divisible by 2, and so are not prime.

• All odd numbers terminating with 3 or 9 are divisible by 3, and so are not prime.

• There exist prime numbers of two or more digits which terminate with 1, 5, 7 or 8

Hence, the set of natural numbers terminating with 1, 5, 7 or 8 must contain all prime

numbers greater than 3, and excludes all odd numbers divisible by 3.

It follows that this set is the minimum set to contain all primes greater than 3.

Re-arranging the terminal digits as 5, 7 and 8, 1 shows the set to be of the form:

(6n ± 1)

Therefore, the minimum set of natural numbers to contain all primes is:

{2, 3, (6n ± 1)} n Œ N

This last statement is a factual property of prime numbers and is therefore true regard-

less of the number-base. it also explains the occurrence of ‘twin primes’, since the only possible

positions for primes greater than 3 are ’each side’of the multiples of 6.

The fact that prime-number positions are completely controlled by 6 (itself the product of

2 and 3, and the companion of our dozenal base) is often not realized even by those with an inter-

est in the subject. It is never found in school text-books, and even Hall & Knight do not mention

it in their ‘Higher Algebra’, which is regarded as a standard work.

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Prime Numbers - Base Patterns Page 3