Definition of a composite number

[jpanknin]

I'm writing a proof for a discrete math class that is asking for proof that an integer is not prime (that for a perfect square, the integer immediately preceding the perfect square is not prime) where n is the perfect square and [imath]n = m^2[/imath] and the integer immediately preceding n is [imath]m^2 - 1[/imath]. I've factored this into [imath]m^2 - 1 = (m + 1)(m -1)[/imath]. My problem is that the definition in our book is:

An integer n is composite if, and only if, n > 1 and n=rs for some integers r and s with 1 < r < n and 1 < s < n.

My question is: why must r and s be positive (1 < r < n). If r and s are both negative, we still get the same positive n value. So for my problem of (m + 1)(m - 1), theoretically these could both be negative and still satisfy [imath]m^2 - 1 = (m + 1)(m - 1)[/imath]. For example, [imath]9 = 3^2 = (-3)^2[/imath] and [imath](-3 + 1)(-3 - 1) = (-2)(-4) = 8[/imath], which satisfies the problem. Is there a constraint that the factors of a composite number must be positive?

 

[Dr.Peterson]

You might answer your own question by considering whether, if a number has negative factors, that implies that it also has positive factors, and thereby satisfies the definition of a composite number.

That is, the definition is sufficient without explicitly mentioning negative factors. It is not necessary to consider negative factors, because that doesn't change the result, and only makes things more complicated to think about..

On the other hand, since they are only allowing composite numbers to be positive, it would appear that they are simply not interested in negative numbers in this context, and just ignore them. The concept can be extended to negative integers (and negative factors), but this aspect of number theory focuses on positive integers.

 

[fresh_42]

The problem with [imath] \pm 1 [/imath] is that they are the units. So everything we say about divisibility is always only up to units (invertible elements). I would define a composite integer [imath] a [/imath] as an integer for which there is a prime [imath] p\neq \pm a [/imath] such that [imath] p\,|\,a. [/imath]

Primes and irreducible numbers are the same for integers, and a composite number is a reducible number, i.e., a non-irreducible number, i.e., a non-prime integer.