Prime Numbers in Different Bases
Date: 10/07/98 at 21:18:07 From: Jorge Diaz Subject: Prime numbers Hi, Dr. Math, Here is a question I have for you. It's on prime numbers. Are all prime numbers the same in all bases? If 21 is a prime, are 10101 (in binary), and 15 (in hexadecimal) also primes? I'm taking a course in Assembly Language Programming, and I was wondering if primality as such is related at all to the number system I am using? What would happen, for instance, if I chose as a base a prime number, such as thirteen? Also, I've recently begun to study fractal geometry, a subject which, along with chaos theory, I find fascinating. I have a strong suspicion that the distribution of prime numbers might be related to fractal geometry, and if I had enough time I would perhaps pursue this subject. By taking a peek at your files, I was able to gather that there is no polynomial function f(n) that will give a list of all the primes, but I was wondering if log f(n) might, if f(n) were at the limit of an infinite interation of the same function, or the limit of an infinite composition of a function with itself, iff, that is, both things are otherwise equal. I would appreciate any insight you might have on this matter. Thank you. Jorge Diaz
Date: 10/07/98 at 23:34:01 From: Doctor Mike Subject: Re: Prime numbers Hi Jorge, A prime is a prime no matter which base you use to represent it. On the surface one might think that in Hex you would have 3*5 = 15 as "usual," but it really turns out that 3*5 = F. The example 21 doesn't work too well because it is not prime. The base ten number 37 is better, because it is prime, but its Hex representation is 25, which sort of looks non-prime. Hex 25 is not, however, repeat not, 5 squared. Okay, enough for examples. The fact of being prime or composite is just a property of the number itself, regardless of the way you write it. 15 and F and Roman numeral XV all mean the number, which is 3 times 5, so it is composite. That is the way it is for all numbers, in the sense that if a base ten number N has factors, you can represent those factors in Hex and their product will be the number N in Hex. Relating to your question about base 13, the base ten number 13 will be represented as "10" in that system, but "10" will still be a prime, because you cannot find two numbers other than 1 and "10" that will multiply together to make "10". I hope this helps you think about primes in other bases. I don't have any insight into fractal geometry and primes, except to say that you will probably be wise to pursue your interest in chaos and fractals. There may be something in there of interest for you to discover. - Doctor Mike, The Math Forum http://mathforum.org/dr.math/
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