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I am studying to become a mathematics teacher in the Netherlands. I recently got this question on an exam and I wonder whether this question is strictly speaking well-written:

Assume that $p_1$ and $p_2$ are twin primes with $p_1 \geq 5$ and $p_2 = p_1 + 2.$ Show that you can write $p_1$ and $p_2$ as $p_1 = 6n - 1$ and $p_2 = 6n + 1$ (with $n \in \Bbb N).$

The two formulas given for $p_1$ and $p_2$ produce all twin primes with $p_1$ equal or bigger then $5,$ but they additionally produce pairs of numbers that are not even both primes. The fact that the formulas produce more then just twin primes mean to me that the twin primes can't be written as those formulas. So this works in one direction, but not the opposite direction. But to me the question seems to imply that this will work in both directions: where twin primes $p_1$ and $p_2$ can be written as, or defined by, those formulas.

The key of the argument with my 'professor' lies in the "(with $n \in \Bbb N$)" part. He says that the twin primes being expressible with those two formulas with $n \in \Bbb N$ doesn't mean that this holds for every $n \in \Bbb N$.

To me, the question is strangely formulated, and possibly incorrectly formulated. Is the formulation of the question correct or not?

ryang
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Sutaner
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    For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – Henrik supports the community Feb 07 '25 at 21:08
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    The formulation looks fine to me. I don't think "(with $n\in\mathbb N$)" implies that it holds for all $n\in\mathbb N$. But you're saying that you're studying to become a teacher in the Netherlands, does that mean the original question was in Dutch? Then we're just talking about your translation. If that's the case you should probably find some mathematicians who speak Dutch and ask them. – Henrik supports the community Feb 07 '25 at 21:16
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    You're being asked to show that all twin primes (except 3 and 5) have the form $6n \pm 1$. That is a different statement from the converse, "All integer pairs of the form $6n \pm 1$ are twin primes”, which is not true (the smallest counterexample being $n=4$). – Dan Feb 07 '25 at 21:34
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    You're overcomplicating the question. This is like someone ask you to prove that any prime number $p$ greater than $2$ is odd, and you make a complicated reflexion about how not every odd number is prime. – jjagmath Feb 07 '25 at 21:53
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    Though the question arose in a simple number theory problem it is really a question about logical inference and has nothing to do with number theory, so I have removed that tag. – Bill Dubuque Feb 07 '25 at 23:48
  • And if you want an answer to the number theory problem, it's already on this site. – Dan Feb 08 '25 at 00:15
  • https://math.stackexchange.com/questions/41623/is-that-true-that-all-the-prime-numbers-are-of-the-form-6m-pm-1 Yes, this proves that primes are of the form $6n + 1$ and also proves that twin primes can be written in the form $6n\pm1$. But it is true for all prime numbers other than 2 and 3, hence the condition that $p_1\ge5$ – ThankYouForFlyingRyanair Feb 08 '25 at 10:37

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The question is fine as written. It says that if you have a pair of twin primes greater than or equal to $5$, they can be written in the form $6n \pm 1$. In other words, it says that all pairs of twin primes can be written in that form. It does not say (or ask you to prove) that all number pairs of that form are in fact twin primes, which (as you know) is false.

In other words, being able to write a pair of numbers greater than or equal to $5$ in the form $6n-1, 6n+1$ for some $n \in \Bbb N$ is a necessary but not sufficient condition for the pair of numbers to be twin primes. That's what the question says and that's also what's true.

Robert Shore
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    Thank you so much for that clear answer with clear explanation! That's what I needed to hear to understand it. – Sutaner Feb 07 '25 at 21:50
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In the given question, the word "with" appears twice, playing distinct roles.

  1. Assume that $p_1$ and $p_2$ are twin primes with $p_1 \geq 5$ and $p_2 = p_1 + 2.$

    Here, the word "with" is introducing a condition (stipulating a restriction on / subset of the twin primes), and its logical sense is AND:

    • Assume that $p_1$ and $p_2$ are twin primes such that $p_1 \geq 5$ and $p_2 = p_1 + 2.$
    • Assume that $p_1$ and $p_2$ are twin primes and that $p_1 \geq 5$ and $p_2 = p_1 + 2.$

  2. Show that you can write $p_1$ and $p_2$ as $p_1 = 6n - 1$ and $p_2 = 6n + 1$ (with $n {\in} \Bbb N$).

    Here, on the other hand, the word "with" is specifying existential quantification $(\exists)$:

    • Show that for some $n \in \Bbb N,$ it holds that $p_1 = 6n - 1$ and $p_2 = 6n + 1.$

All in all, the question is asking for the proof that for every pair of twin primes $p_1$ and $p_2,$ $$(p_1 \geq 5\,\text{ and }\,p_2 = p_1 + 2)\implies\exists n {\in} \Bbb N\,(p_1 = 6n - 1\,\text{ and }\,p_2 = 6n + 1).$$

ryang
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You seem to be committing the logical fallacy of affirming the consequent.

Let $P$ denote the proposition “$p_1 \ge 5$ and $p_2 = p_1 + 2$ are both prime numbers.”

Let $Q$ denote the proposition “$\exists n\in\mathbb{N}: p_1 = 6n-1, p_2 = 6n+1$”.

The exam question asks you to show that $P \implies Q$, which is true.

It does not ask you to show that $Q \implies P$, which is false. A counterexample is $n = 4$, $p_2 = 25 = 5^2$. A double counterexample is $n=20$, $p_1 = 119 = 7 \times 17$, $p_2 = 121 = 11^2$.

The question is correctly formulated. Just make sure you pay attention to which direction the propositional arrow points. Perhaps some confusion was introduced by a Dutch-English translation issue.

Dan
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