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I asked Chat GPT to suggest a number theoretic conjecture. It came up with the following interesting conjecture:

Conjecture (Chat GPT): For each even natural number $n$, there is a prime $p$ such that $p+n^2$ is also prime.

(I am not sure it stated even but this is clearly necessary, e.g., take $n=5$.)

Equivalently, if $P$ is the set of primes, then the difference set $P-P$ contains all squares of even natural numbers.

Is this conjecture true or false?

What is known about variations of this conjecture, with some other function $f(n)$ instead of $n$?

I am not a number theorist, but every mathematician finds number theoretic questions interesting.

Boaz Tsaban
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    It is certainly true and follows from standard conjectures, but widely open even for $n=2$. – Fedor Petrov May 22 '23 at 09:38
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    @FedorPetrov actually it only asks for one prime $p$ so that $p+n^2$ is prime, not infinitely many (which is the case for the twin primes conjecture/bounded gaps theorem). So the $n=2$ case is very much solved! – R. van Dobben de Bruyn May 22 '23 at 09:45
  • @FedorPetrov Do you mean that there is a conjecture that for each k there is an arithmetic progression of k primes with difference 2? – Boaz Tsaban May 22 '23 at 09:47
  • @R.vanDobbendeBruyn Does the set P-P have full density? (Density 1) – Boaz Tsaban May 22 '23 at 09:49
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    No arithmetic progression of course (look modulo 3). Just infinitely many pairs. – Fedor Petrov May 22 '23 at 09:51
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    @BoazTsaban Conjecturally P-P contains every even number. More generally, Hardy and Littlewood conjectured that for every 'admissible' tuple $(h_1,\ldots,h_k)$ of positive integers there are infinitely many positive integers $m$ such that ${m+h_i}{i=1}^{k}$ are simultaneously prime (and they predict the count of these $m$ up to $x$, asymptotically). Your conjecture relates to $h_1=0,h_2=n^2$. 'Admissible' means there is no prime $p$ such that ${ h_i \bmod p}{i} = \mathbb{Z}/p\mathbb{Z}$ (see Fedor Petrov's comment). For more information see https://en.wikipedia.org/wiki/Prime_k-tuple . – Ofir Gorodetsky May 22 '23 at 09:52
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    @R.vanDobbendeBruyn you are correct of course. But I am afraid that still this is open for large specific $n$. – Fedor Petrov May 22 '23 at 09:53
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    Going off the tangent a bit, is there any specific infinite set, not defined using the primes implicitly or explicitly (I know I'm being a bit vague here), known to be contained in $\mathbb{P}-\mathbb{P}$? – Yaakov Baruch May 22 '23 at 10:58
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    @YaakovBaruch Not that I am aware of, but there are partial results about the density of P-P and its structure, see https://link.springer.com/chapter/10.1007/978-3-319-22240-0_10 and https://www.ams.org/journals/proc/2017-145-09/S0002-9939-2017-13533-3/ – Ofir Gorodetsky May 22 '23 at 11:19
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    in the spirit of the question, I asked chatGPT 4 if this conjecture was known to be true/false or open. It answered: "Up until my last training data in 2021, this conjecture was not resolved. For the most current and accurate information, you should refer to recent mathematical literature or ask a professional mathematician or number theorist." – Carlo Beenakker May 22 '23 at 12:11
  • Maybe we can ask ChatGPT to prove the conjecture? – JRN May 22 '23 at 13:50
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    I would be much happier if all mention of GPT were removed from the original question. – Yemon Choi May 23 '23 at 02:07
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    @Yemon I agree, it adds nothing to the mathematics. – David Roberts May 23 '23 at 09:16
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    @YemonChoi I disagree. This conjecture was made by Chat GPT, and removing its mention would be plagiarizm. – Boaz Tsaban May 23 '23 at 15:24
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    @BoazTsaban to me, this is like saying "I rolled some dice and I used the results as coefficients for some polynomial, what is the Galois group of this polynomial". LLMs by their nature cannot understand their output, so saying this is a "conjecture by ChatGPT" seems to be attributing agency/sentience where it doesn't exist. – Yemon Choi May 23 '23 at 15:31
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    @YemonChoi This is yet to be discussed for some decades... it is still unclear how far Chat GPT is from human intelligence. I would postpone the decision for at least several years. – Boaz Tsaban May 23 '23 at 19:49
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    @BoazTsaban chatGPT is nothing else but a text generator generating the most probable sequel of a sequence of words. You can play with the openAI playground to see that. Hence all hallucinations that this so-called AI has: it definitively cannot be trusted. When the sequence of words looks like a question, the most probable sequel is a sequence of words which looks for us, humans, like an answer because of its training. chatGPT does not understand what it's talking about. – Philippe Gaucher May 24 '23 at 07:57
  • @PhilippeGaucher This discussion goes far... This is too subtle an issue to be dealt with in short discussions. Time will tell, I guess. – Boaz Tsaban May 24 '23 at 11:10
  • @BoazTsaban Try https://platform.openai.com/playground. It is interesting to play with it to understand how it works; keep in mind that for more recent chat bots, the discussion starts with a hidden pre-prompt which is supposed to give some rules of behaviour. – Philippe Gaucher May 26 '23 at 06:02

2 Answers2

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The more natural conjecture, that every even number arises as the difference of two primes, was asked on math.SE years ago. That question remains open, and I have a hard time imagining that restricting to squares makes the problem any more tractable.

EDIT: The conjecture that every even number is the difference of two primes is apparently sometimes called Maillet's conjecture.

Timothy Chow
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    I believe this conjecture predates the coming of stackexchange by many decades. Already in 1849 de Polignac conjectured that every even number arises infinitely often as the difference of consecutive primes. – Gerry Myerson May 23 '23 at 03:19
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    @GerryMyerson Good point. On the other hand, conceivably the question of whether every even number arises at least once as the difference of two not-necessarily-consecutive primes could be easier than Polignac's conjecture. – Timothy Chow May 23 '23 at 03:22
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Ancient Greeks conjectured that there are infinitely many pairs of primes which differ by 2 (twin primes). A natural widely believed generalization is that 2 may be replaced by every even number. Moreover, it is expected that for every integer $T>0$ there are about $C n/\log^2 n$ numbers $p\leqslant n$ for which $p$ and $p+2T$ are both primes.

What is proved is that there are infinitely many pairs of primes with the same difference (Zhang and beyond). But it is not known for difference 2 or 4. If you (or chatgpt) need only one pair, this is certainly checked for all not too large integers, but I am afraid that still open for large enough integers.

Fedor Petrov
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    The question asks if there is one pair of primes $(p,p+4)$, not infinitely many. – R. van Dobben de Bruyn May 22 '23 at 09:50
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    Does $7+ 2^2= 11$ count? – user25406 May 22 '23 at 18:04
  • Can you provide a citation for the ancient Greek conjecture? – Gerald Edgar Jun 03 '23 at 01:41
  • @GeraldEdgar well, I was sure in this as a common knowledge, but now I am less certain. Greeks are mentioned in several places (see below, for example), but without citation and assurance. https://www.nature.com/articles/nature.2013.12989 – Fedor Petrov Jun 03 '23 at 05:09
  • About twin primes, the Nature snippet says, in passing, "Some attribute the conjecture to the Greek mathematician Euclid of Alexandria". That seems not convincing. Euclid did not even say "there are infinitely many primes"; he said "Given a [finite] list of primes, to construct a prime not in the list." – Gerald Edgar Jun 03 '23 at 11:58
  • @GeraldEdgar well, it is essentially the same, up to logical nuances. We often rephrase ancient theorems in modern language, like "square root of 2 is irrational" instead of "the side and the diagonal of the square do not have a common measurament". – Fedor Petrov Jun 03 '23 at 12:49