Online Course Discussion Forum
II-A Number Theory 3.28
Shouldn’t we show that the condition that the two odd primes must be consecutive is unnecessary? However, the question let us show that it is necessary.
By "necessary" here we mean that we want to help explore why the statement:
- "If p and q are consecutive odd primes, then p+q has at least 3 factors in its prime factorization" is always true,
but the statement
- "If p and q are odd primes, then p+q has at least 3 factors in its prime factorization" is NOT always true.
Hence, it's necessary that we have consecutive primes for this statement to be always true.
Hope this helps!
So we can have one positive and one negative prime number they are not consecutive? E.g. -3 and 5 sums 2 which only has two factors. Is this a counterexample?
Good question!
Typically a prime number is defined as an integer greater than $1$ with only $1$ and itself as factors. Hence, usually we wouldn't talk about a number like $-3$ being prime or not being prime.
Thus, while you are correct that $-3+5=2$ has only two factors, we wouldn't consider $-3$ a prime number.
If we cannot choose a negative integer, then why is the statement below false? What is a counterexample?
- "If p and q are odd primes, then p+q has at least 3 factors in its prime factorization" is NOT always true.
Sorry for the confusion, let me try to summarize the results a little bit for clarity.
- The statement "If p and q are consecutive odd primes, then p+q has at least 3 factors in its prime factorization" is true (for any values of p and q). This is Example 3.8.
- If we remove the word "consecutive" to get the statement "If p and q are odd primes, then p+q has at least 3 factors in its prime factorization", then this statement is false for some values of p and q. Of course if p and q are consecutive this is true, but, if p = 3 and q = 7, for example, 3+7=10 = 2*5 only has two primes in it's prime factorization.
Note we are only need one counterexample here. Others do exist, like p = 3 and q = 11 (with p+q=14=2*7), but sometimes there are at least 3 factors in the prime factorization even if p and q are not consecutive. For example, p = 5 and q = 11 with p+q = 16=2*2*2*2 has 4 factors in it's prime factorization.
Hope this helps a little!
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