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I argued it as follows, let $p, q, r$ and $s$ be predicates

$p$: "$m$ is divisible by $k$"

$q$: "$k$ is divisible by $n$ ($n < k$ and $m$ is divisible by $n$) "

$r$: " $k$ is the smallest factor of $m$ other than $1$"

$s$: "$k$ is prime"

The true arguments: $(p \land r), (q\iff\lnot s), (r \implies\lnot q)$

Argument:

$ p \land r $


$ r $

$ r \implies \lnot q $


$ \lnot q$

$q \iff \lnot s$


$s$

$ \therefore p \land r \implies s$ = If $k$ is the smallest factor of any integer $m, k$ is prime.

Is this correct?

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    $\begingroup$ $1$ is the smallest divisor of every natural number. $\endgroup$ – Adam Hughes Aug 1 '14 at 5:55
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    $\begingroup$ Well $1$ would be the least positive divisor. In general for a natural number $n$, $-n$ would be the least divisor. $\endgroup$ – Peter Woolfitt Aug 1 '14 at 6:05
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Consider the inverse: that there exists an integer $m$ where the smallest factor $n$ is not prime. If that is the case then there exists a prime integer $k$ such that $k < n$ and $k$ divides $n$, thus $k$ also divides $m$, which is a contradiction.

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I think there should be some more precision regarding definitions. For instance, the statement "$p$ is prime" is equivalent to saying:

For every pair $a,b$ of integers such that $p$ divides $ab$, one of $a$ or $b$ is divisible by $p$.

I don't see how this was involved in your proof. Alternatively, you could use the definition of irreducibility and apply the prime factorization theorem to get a proof closer to the one you are suggesting. In this case, I think you are close, except you need to add existential quantifiers to some of your statements.

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Not really a good "strict" proof, if you ask me. For example, you just assumed that $r\implies \neg q$ is true, I see no reason why I should (strictly speaking) believe that.

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