Prove that if $p, p+2$ and $p+4$ are primes, then $p=3$

Question

Since $p+2$ and $p + 4$ are prime, $3$ doesn't divide either of them. But then $3 \mid p+3$, which implies $3 \mid p$. Since $p$ is prime, then $p = 3$.

Is it this simple?

Your solution is correct. I would only add that $p+2$ and $p+4$ are primes which are bigger than $3$ (it is obvious but still should be written), this is why $3$ doesn't divide them. — Mark, Oct 25 '18 at 22:41
If I was teaching an undergraduate introduction to proofs, I would ask for some justification of the "But then" assertion. As written, it sort of sounds like you're saying that if $3$ doesn't divide two numbers, then it does divide their average.... — Barry Cipra, Oct 25 '18 at 22:47
@BarryCipra I thought it was obvious that I was using that $3$ divides any three consecutive numbers, but I see how that is a valid concern. Thanks. — Matheus Andrade, Oct 25 '18 at 22:48
When first learning to write proofs, it helps to be super explcit, even if it makes your proofs tedious and verbose. (If the grader doesn't like it, that's their tough luck.) As you get more experienced, you get a feel for which details are crucial and which ones can be skipped over. — Barry Cipra, Oct 25 '18 at 22:56

score 3 · Answer 1 · answered Oct 25 '18 at 22:37

3

Hint $ $ They're $\,\equiv p,\, p\!+\!1,\, p\!+\!2\pmod 3\,$ so one is divisible by $3$

answered Oct 25 '18 at 22:37

Bill Dubuque

282,220

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Isn't that sort of exactly what I wrote? – Matheus Andrade Oct 25 '18 at 22:39
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@Matheus The hinted ideas is to use that a sequence of $n$ consecutive integers contains a multiple of $n$ or, equivalently, forms a complete set sof residues (remainders). It's not clear if that's what you are using. – Bill Dubuque Oct 25 '18 at 22:44
I see now. Thanks! – Matheus Andrade Oct 25 '18 at 22:46
See here for a proof of said claim about consecutive integers (and links to generalizations). $\ \ $ – Bill Dubuque Sep 28 '24 at 23:11

score 2 · Accepted Answer · answered Oct 25 '18 at 22:40

2

$3$ has to divide exactly one of $p, p+2, p+4$ (from simple congruency conditions). Since these are assumed to be primes, this implies $3$ IS one of $p, p+2, p+4$. Checking the $3$ conditions, this leads to $p=3$.

answered Oct 25 '18 at 22:40

AntoineJW

38
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Thanks. I'm accepting your answer and upvoting you too since you're a new contributor. – Matheus Andrade Oct 25 '18 at 22:42
That's great. Thanks! – AntoineJW Oct 25 '18 at 22:43

Prove that if $p, p+2$ and $p+4$ are primes, then $p=3$

2 Answers2