Disproof: there exists an integer $k ≥ 4$ such that $2k^2 − 5k + 2$ is prime.

Question

Show that the following statement is false: There exists an integer $k ≥ 4$ such that $2k^2 − 5k + 2$ is prime.

Assume this is true:

For all integers $k ≥ 4$, $2k^2 − 5k + 2$ is not prime.

Proof:

Suppose $k ∈ Z$ such that $k ≥ 4$. By definition of prime, $n=rs$ for some integer $r$ and $s$, where $r = 1$ and $s = n$ or $r = n$ and $r = 1$. We want to show that $2k^2 − 5k + 2$ is not prime. Then

$2k^2- 5k + 2 ≠rs$

$(2k - 1) (k-2) ≠ rs$

Notice that $2k-1>1$ and $k-2>1$, because $k ≥ 4$. Also, $2k-1$ and $k-2$ are less than $2k^2- 5k + 2$, which means it’s a product of two smaller integers. Therefore, $2k^2 − 5k + 2$ is not prime, which is a contradiction to original theorem.∎

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My first question is whether the definition of prime is needed when contradicting the statement. I thought that maybe the definition is not important, since it says:

$2k^2 − 5k + 2$ is not prime.

Another question is whether my concluding paragraph is too wordy.

This my first time doing a proof by contradiction.

Answer 1

Show that the following statement is false: There exists an integer $k ≥ 4$ such that $2k^2 − 5k + 2$ is prime.

Assume this is true:

For all integers $k ≥ 4$, $2k^2 − 5k + 2$ is not prime.

Proof:

Suppose $k ∈ Z$ such that $k ≥ 4$. By definition of prime, $n=rs$ for some integer $r$ and $s$, where $r = 1$ and $s = n$ or $r = n$ and $r = 1$. We want to show that $2k^2 − 5k + 2$ is not prime. Then

$2k^2- 5k + 2 ≠rs$

$(2k - 1) (k-2) ≠ rs$

Notice that $2k-1>1$ and $k-2>1$, because $k ≥ 4$. Also, $2k-1$ and $k-2$ are less than $2k^2- 5k + 2$, which means it’s a product of two smaller integers. Therefore, $2k^2 − 5k + 2$ is not prime, which is a contradiction to original theorem.∎

This my first time doing a proof by contradiction.

You repeatedly mix up "we want to show blah blah" and "we assume blah blah", making your proof illogical, and confusing to read. For example, despite your boldfaced opening heading, your proof's opening section is actually telegraphing what it wants to show rather than making an assumption (to assume something means to suppose, without any proof, that it is true; this supposition to be subsequently utilised in the proof).

It is clear that your proof is attempting to show that the given statement's negation is true, that is, that the statement itself is false; in other words, it is attempting to disprove the given statement. But a proof of a negation (what you are actually doing) is conceptually different from a proof by contradiction (what you think you are doing)!

• When proving a statement by contradiction, you start by assuming its negation, then derive some contradiction (not necessarily involving the given statement or its negation), then conclude that the given statement must be true.

• When disproving a statement by contradiction, you start by assuming the given statement, then derive some contradiction (not necessarily involving the given statement or its negation), then conclude that the given statement must be false.

For this exercise, proof by contradiction is unnecessary.

Answer 2

Prove the following claim is true: "$P$"

Prove by contradiction, assume "$\neg P$" is true.

Prove the following claim is false: "$Q$"

Prove by contradiction, assume "$\neg Q$" is false $\Leftrightarrow$ assume "$Q$" is true.

In your case:

show that the following statement is false:

There exists an integer $k ≥ 4$ such that $2k^2 − 5k + 2$ is prime.

$Q:$ "There exists an integer $k ≥ 4$ such that $2k^2 − 5k + 2$ is prime."

So you should assume $Q$ is true, namely

Assume "There exists an integer $k ≥ 4$ such that $2k^2 − 5k + 2$ is prime." is true.