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My question is about number theory about the prime numbers.

Here is the question:

Prove that there is only one prime trio of the form $p$,$p+2$,$p+4$.

I mean,we can only find the primes 3,5,7

That satisfies the condition.

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Let $n$ be an integer. Then one of $n$, $n+2$, $n+4$ is divisible by $3$.

For the remainder when $n$ is divided by $3$ is $0$, $1$, or $2$.

If it is $0$, then $n$ is divisible by $3$.

If it is $1$, then $n+2$ is divisible by $3$.

If it is $2$, then $n+4$ is divisible by $3$.

A number $\gt 3$ which is divisible by $3$ cannot be prime.

Now suppose that $n,n+2,n+4$ are all prime.

Then $n$ cannot be $1$, since $1$ is not prime.

Also, $n$ cannot be $2$, since $4$ (and $6$) are not prime.

Certainly $n$ can be $3$, giving the primes $3,5,7$ of the question.

And $n$ cannot be greater than $3$, for then all of $n,n+2,n+4$ are $\gt 3$, and one of them is divisible by $3$, so not prime.

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