# Number of positive integers n such that 3n - 4, 4n - 5, 5n - 3 are all prime

I couldn't find an answer to this but I did find a proof that this question is incorrect:

We have $$3n - 4$$ = prime, so, n cannot be of the form $$2q$$, $$4q$$, $$8q$$, because then I'd be able to factorise it

Similarly, $$4n - 5$$ = prime, so n is not of the form $$5q$$

And, $$5n - 3$$ = prime, so n is not of the form $$3q$$, $$6q$$, $$9q$$

Which means, $$n$$ is certainly an odd prime.

If we look at $$5n - 3$$, $$5$$ and $$n$$ are both odd, so their product is odd

$$3$$, when subtracted from an odd number, would always give an even number, which creates a contradiction.

So who's wrong, me or the problem? If I'm wrong, do tell me why and also the solution to the problem, thanks

• What do you mean, "this question is incorrect"? Are you just saying that you think the answer is $0$?
– lulu
Commented Apr 13, 2020 at 15:22
• Just to say, there is an integer $n$ for which all three of those are prime. Your idea are ok, but your conclusion is (slightly) wrong. You conclude that one of the three numbers must be even, which is true, but you forget that there is an even prime.
– lulu
Commented Apr 13, 2020 at 15:23
• 5n - 3 = 2 implies n = 1. Commented Apr 13, 2020 at 16:12
• 4n - 5 would then return a negative value Commented Apr 13, 2020 at 16:13
• As a general suggestion: always work examples. In this case, trial and error would turn up the one good $n$ almost immediately.
– lulu
Commented Apr 13, 2020 at 16:13

Notice that the sum of the three primes is $$(3n-4)+(4n-5)+(5n-3)=12(n-1)$$ which is clearly even. Therefore, not all of the primes are odd. It follows that one of them has to be equal to the only even prime, $$2$$.
Since $$n\in\mathbb{N}$$, either $$3n-4=2$$ or $$5n-3=2$$, implying that either $$n=2$$ or $$n=1$$.
$$n=1$$ is rejected since in this case, $$3n-4<0$$. $$n=2$$ gives $$2, 3, 7$$ as our primes. Therefore, $$n=2$$ is the only solution to the problem.
if $$n \equiv 1 \pmod 6$$ then $$5n - 3 \equiv 2 \pmod 6$$ which is not prime similarly if $$n \equiv -1 \pmod 6$$ then $$5n - 3 \equiv 4 \pmod 6$$ which is also not prime