is "1" considered as a PRIME NUMBER?
It depends on whether the definition of a "prime number" stipulates the syntagma "proper divisors".If so,1 is not a prime number,if not,then it is a prime number.
as long as i know, the definition for prime number is a natural number that is bigger than one and can be divided by only two numbers - itself and one. for example, the number seven is a prime number. it is natural, bigger than 1 and can be divided by itself (7) and one.
Apparently it doesn't,but 1 is still not a prime number.Maybe you should have checked this page first?? :tongue2:
1 is not a prime number
However,the definition is not unique:other definition There's something weird with this page.According to their definition,1 should be a prime,but in the next line,when they give examples,1 is missing...
A nice discussion is here:
thanks guyz :-D
The *current* definition is that 1, as it is a unit (divides 1) is not a prime. This the is current convention. At one point it was considered to be a prime, but then for reasons that I suggest come from studying different systems of arithmetic it came to be easier and clearer for the convention to be that 1 is not prime. These other systesms are properly called rings, and in soem rings there are many other numbers that divide 1, and it was necessary to carefully distinhuish between units and non-units especially in decompositions into primes.
One is not a prime number because it would violate unique prime decomposition. That is 1X1= 1x1x1=1,etc.
But with a prime it only divides a given integer a unique number of times. Thus 54 = (3^3)(2). There is no way that 2 or 3 could divide it a different number of times!
1 AS A PRIME WOULD VIOLATE THE FUNDAMENTAL THEOREM OF ARITHMETIC.
In fact what had been quoted above, even if somewhat ambigious, tells you as much if you read the whole thing: http://odin.mdacc.tmc.edu/~krc/numbers/prime.html
I think you have it backwards. The wording of the fundamental theorem is based upon the convention that 1 is not a prime. "primeness" as with any other definition of mathematics is purely a convention.
Well one of the first organized efforts on primes was the Sieve of Eratosthenes, who lived 276BC-194BC; and the method is to write out a successive list of integers begining with the first prime 2, and when we reach the p prime remaining, to knock out every pth number in the list.
HOWEVER, if one were a prime, we would knock out every number....So the method would not work!
I would love to know where and when any prominent mathematician has ever defined 1 as a prime. Apparently Euclid not consider 1 as a prime, in fact, he excluded it from his definition of a number. (See reference below.)
There is a website where the author says, I understand that a hundred years or so ago, some books actually said that one was a prime number. I've never seen such a book, but I'd love to see it. http://www.mazes.com/primes/one.html
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