Are the prime numbers we know peculiar to the decimal system?

Question

You will now be aware I'm not much of a mathematician asking such an obvious question but I'm interested in this topic. I do know that decimal system primes can be expressed to any other base.

It occurred to me that there is either one prime number, or all numbers are prime in the binary system. As for the hexadecimal system, I'm just as uncertain. Are prime numbers dependent on the base number employed? If they are, why all the interest in the decimal system's primes and not others? My thanks to anyone who can put me right.

Answer 1

Primes are primes, whatever numbering system you use. You can describe 17 buttons in hexadecimal notation, or any other base. It doesn't change the fundamental fact that there are a prime number of buttons which cannot be divided into two equal amounts.



The "primeness" is a more fundamental feature of a number than the notation used to describe it. It just so happens that decimal notation is the most common way of talking about it.

Answer 2

A prime number is a prime number, it makes no difference how you write the numbers down, even Roman numerals, which technically you can't multiply or divide, I don't think there is any particular interest in decimal primes, I think that is the system that virtually everyone uses, even if you use base 12 or any other. Actually I just noticed that the number 13 in base 12 becomes 11 which is also a prime number in base 10. Some things are just cannot be explained satisfactorily, mathematicians have been stumped by primes for centuries.

Answer 3

Arithmetic does not depend at all on the way numbers are represented (using decimal or any other notation). To say that 5*7 = 35 in base 10 is completely equivalent to saying that 101*111 = 10011 in binary. Thus if you compute the prime factorization of any number in any base then the exact same prime factorization holds in any other base. So all bases yield the same notion of prime.



Remember that the binary/decimal/hexadecimal representations of a number are no more than representations; they are not logically relevant to underlying structure of numbers. Changing a representation of a number from base 10 to base 2 is analogous to translating a passage from French to Spanish - it looks different, but the meaning is still the same.

Answer 4

Renoir's and Robert's answers are correct and I don't understand why you complain about them. A prime number is a prime number, full stop. A number base (decimal, for example) is simply a choice of ways of expressing that number. We choose decimal because we have ten fingers and thumbs. Hence the interest in "decimal primes" - because that is the number base most of us use. If a number is prime in decimal then it is also prime in any other base. For instance, the numbers 1011 (binary), B (hexadecimal) and 11 (decimal) are all the same number and are all prime - just expressed in different ways. If you worked in (say) communications theory then you would be interested in prime numbers expressed in other bases - but they are the same numbers.

Answer 5

{blank} = {blank} No input = no output. Simple enough - did you do the reading on how binary works -- each digit is a power of 2 1101 1 x 2^3 = 8 1 x 2^2 = 4 0 x 2^1 = 0 1 x 2^0 = 1 Now add them up 8+4+0+1 = 13

Answer 6

The number base is irrelevent. The number 7 can be expressed in whatever number base you wish. It will always be prime.



Think of it as a pile of apples. Can you divide up a pile of seven apples into an equal number of piles?