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.

 

[jpanknin]

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.

I get this part, but from there how can I then prove what number m actually is? If it can be both, I can't make a definitive statement that it is positive or negative. Or are you saying it doesn't really matter? Or that it can't be proved from the information given?

 

[Dr.Peterson]

I get this part, but from there how can I then prove what number m actually is? If it can be both, I can't make a definitive statement that it is positive or negative. Or are you saying it doesn't really matter? Or that it can't be proved from the information given?

Read the definition carefully!!!

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.

Given that [imath]m=-3[/imath], your number is [imath]n=m^2-1=(-3)^2-1=(-3+1)(-3-1)=(-2)(-4)=8[/imath]. You can't use factors -2 and -4 to demonstrate that it is composite, but [imath](-2)(-4)=(2)(4)[/imath], and you can use that.

The definition doesn't say that every pair of factors must be positive in order to be composite; it says that there must be some such pair. Since you can always change a pair of negative factors to a pair of positive factors, therefore the requirement of positive factors does not affect the meaning of the definition. You still get exactly the same set of composite numbers.

To directly answer the question you asked here, you don't need to "prove what m is"; you are given some m. But you can choose the factors so that they fit the definition, either [imath](m+1)(m-1)[/imath] or [imath](-m+1)(-m-1)[/imath]. One of those will be a pair of positive factors.

Or, if you prefer, you can say that since [imath]m^2=(-m)^2[/imath], if you are given a negative m, you can replace it with [imath]-m[/imath], and the question is still about the same number [imath]m^2-1=(-m)^2-1[/imath].

Now, if you wish, show us the exact statement you are supposed to prove, and show us an attempt at the proof, using these ideas. We can talk more concretely when you do that.