# 3.4: Disproofs

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The idea of a “disproof” is really just semantics – in order to disprove a statement we need to prove its negation.

So far we’ve been discussing proofs quite a bit, but have paid very little attention to a really huge issue. If the statements we are attempting to prove are false, no proof is ever going to be possible. Really, a prerequisite to developing a facility with proofs is developing a good “lie detector.” We need to be able to guess, or quickly ascertain, whether a statement is true or false. If we are given a universally quantified statement the first thing to do is try it out for some random elements of the universe we’re working in. If we happen across a value that satisfies the statement’s hypotheses but doesn’t satisfy the conclusion, we’ve found what is known as a counterexample.

Consider the following statement about integers and divisibility:

##### Conjecture $$\PageIndex{1}$$

$∀a, b, c ∈ \mathbb{Z}, a| bc \implies a| b ∨ a| c.$

This is phrased as a UCS, so the hypothesis is clear, we’re looking for three integers so that the first divides the product of the other two. In the following table, we have collected several values for $$a$$, $$b$$ and $$c$$ such that $$a| bc$$.

$$a$$ $$b$$ $$c$$ $$a| b ∨ a| c ?$$
$$2$$ $$7$$ $$6$$ $$\text{yes}$$
$$2$$ $$4$$ $$5$$ $$\text{yes}$$
$$3$$ $$12$$ $$11$$ $$\text{yes}$$
$$3$$ $$5$$ $$15$$ $$\text{yes}$$
$$5$$ $$4$$ $$15$$ $$\text{yes}$$
$$5$$ $$10$$ $$3$$ $$\text{yes}$$
$$7$$ $$2$$ $$14$$ $$\text{yes}$$
##### Practice

As noted in Section 1.2 the statement above is related to whether or not a is prime. Note that in the table, only prime values of $$a$$ appear. This is a rather broad hint. Find a counterexample to Conjecture $$3.4.1$$.

There can be times when the search for a counterexample starts to feel really futile. Would you think it likely that a statement about natural numbers could be true for (more than) the first $$50$$ numbers a yet still be false?

##### Conjecture $$\PageIndex{2}$$

$$∀n ∈ \mathbb{Z} + n^2 − 79n + 1601$$ is prime.

##### Practice

Find a counterexample to Conjecture $$3.4.2$$

Hidden within Euclid’s proof of the infinitude of the primes is a sequence. Recall that in the proof we deduced a contradiction by considering the number $$N$$ defined by

$N = 1 + \prod_{k=1}^{n} p_k.$

Define a sequence by

$N_n = 1 + \prod_{k=1}^{n} p_k,$

where $$\{p_1, p_2, . . . , p_n\}$$ are the actual first $$n$$ primes. The first several values of this sequence are:

$$n$$ $$N_n$$
$$1$$ $$1 + (2) = 3$$
$$2$$ $$1 + (2 · 3) = 7$$
$$3$$ $$1 + (2 · 3 · 5) = 31$$
$$4$$ $$1 + (2 · 3 · 5 · 7) = 211$$
$$5$$ $$1 + (2 · 3 · 5 · 7 · 11) = 2311$$
$$⋮$$ $$⋮$$

Now, in the proof, we deduced a contradiction by noting that $$N_n$$ is much larger than $$p_n$$, so if $$p_n$$ is the largest prime it follows that $$N_n$$ can’t be prime – but what really appears to be the case (just look at that table!) is that $$N_n$$ actually is prime for all $$n$$.

##### Practice

Find a counterexample to the conjecture that $$1 + \prod^{n}_{k=1} p_k$$ is itself always a prime.

## Exercises:

##### Exercise $$\PageIndex{1}$$

Find a polynomial that assumes only prime values for a reasonably large range of inputs.

##### Exercise $$\PageIndex{2}$$

Find a counterexample to Conjecture $$3.4.1$$ using only powers of $$2$$.

##### Exercise $$\PageIndex{3}$$

The alternating sum of factorials provides an interesting example of a sequence of integers.

$$1! = 1 \\ 2! − 1! = 1 \\ 3! − 2! + 1! = 5 \\ 4! − 3! + 2! − 1! = 19 \\ \text{et cetera}$$

Are they all prime? (After the first two $$1$$’s.)

##### Exercise $$\PageIndex{4}$$

It has been conjectured that whenever $$p$$ is prime, $$2^p − 1$$ is also prime. Find a minimal counterexample.

##### Exercise $$\PageIndex{5}$$

True or false: The sum of any two irrational numbers is irrational. Prove your answer.

##### Exercise $$\PageIndex{6}$$

True or false: There are two irrational numbers whose sum is rational. Prove your answer.

##### Exercise $$\PageIndex{7}$$

True or false: The product of any two irrational numbers is irrational. Prove your answer.

##### Exercise $$\PageIndex{8}$$

True or false: There are two irrational numbers whose product is rational. Prove your answer.

##### Exercise $$\PageIndex{9}$$

True or false: Whenever an integer n is a divisor of the square of an integer, $$m^2$$, it follows that $$n$$ is a divisor of $$m$$ as well. (In symbols, $$∀n ∈ \mathbb{Z}, ∀m ∈ \mathbb{Z}, n | m^2 \implies n | m.)$$ Prove your answer.

##### Exercise $$\PageIndex{10}$$

In an exercise in Section 3.2, we proved that the quadratic equation $$ax^2 + bx + c = 0$$ has two solutions if $$ac < 0$$. Find a counterexample which shows that this implication cannot be replaced with a biconditional.

This page titled 3.4: Disproofs is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Joseph Fields.