01-15-2008, 07:19 PM
01-15-2008, 07:20 PM
Sorry! the poll is wrong. Here's it properly
01-22-2008, 05:08 AM
B"H
Fascinating subject. 1 is not thought of as a prime these days. However, it is a paradoxical subject. One can only be divided by one and itself. However, it is also true that 1 can be divided by itself and 1 squared. And, for that matter, it can be divided by itself and 1 cubed. So on and so forth. We do not generally think of primes as being divided by themselves AND powers of themself. However, if we DO count 1 as a prime, we have the paradox of having every prime number being divisible by itself and another prime number (1). That would defeat the whole purpose of calling it a "prime." All numbers would be, in essence, composite. In fact, one would be composite. This is too much of a logical paradox, so it is best to leave 1 alone, neither as prime nor composite.
Also, the Sieve of Eratosthenes would be kind of off kilter if we considered 1 a prime. On this Sieve, you eliminate all multiples of any prime to have only primes left. However, the Sieve MUST start with 2 in order to work. If it starts with 1, all numbers above 1 automatically become composite. I say that modern mathematicians have it right---2 is the first prime number. However, I respect 1 enough to leave it alone, neither declaring it prime nor composite. It is, after all, the first natural number.
That's my take.
Fascinating subject. 1 is not thought of as a prime these days. However, it is a paradoxical subject. One can only be divided by one and itself. However, it is also true that 1 can be divided by itself and 1 squared. And, for that matter, it can be divided by itself and 1 cubed. So on and so forth. We do not generally think of primes as being divided by themselves AND powers of themself. However, if we DO count 1 as a prime, we have the paradox of having every prime number being divisible by itself and another prime number (1). That would defeat the whole purpose of calling it a "prime." All numbers would be, in essence, composite. In fact, one would be composite. This is too much of a logical paradox, so it is best to leave 1 alone, neither as prime nor composite.
Also, the Sieve of Eratosthenes would be kind of off kilter if we considered 1 a prime. On this Sieve, you eliminate all multiples of any prime to have only primes left. However, the Sieve MUST start with 2 in order to work. If it starts with 1, all numbers above 1 automatically become composite. I say that modern mathematicians have it right---2 is the first prime number. However, I respect 1 enough to leave it alone, neither declaring it prime nor composite. It is, after all, the first natural number.
That's my take.