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$$P^3 - (P-1)^2=q$$ $P$: prime number

$q$:prime number

examples

$$2^3 - 1^2=7$$

$$3^3 - 2^2=23$$

$$5^3 - 4^2=109$$

$$7^3 - 6^2=307$$

$$11^3 - 10^2=1231$$ This function certainly does not work with all prime numbers, but check even in advanced numbers, my questions:

  1. Could it be a mathematical conjecture that there are an infinite number of prime numbers written in this form?

  2. Is it possible to use this function mathematically?

Mohammed
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    For your first question: yes, it is legitimate to ask such a question. However, this is one of those cases where it is immensely more difficult to answer questions than ask them. We don't even know if there are infinitely many prime numbers $q$ of the form $n^2+1$. (See: https://math.stackexchange.com/questions/44126/primes-of-the-form-n21-hard .) Thus, to be taken seriously, there must be some underlying motivation for picking that particular formula. –  Nov 18 '21 at 14:00
  • Having said that, not all questions are alike, and who knows - maybe yours has an easy answer one way or another. Maybe someone on MSE would know?! –  Nov 18 '21 at 14:00
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    There are many such functions. The key is to make sure the expression is designed such that its never divisible by the first few primes (like $2,3,5$ in our case). If so then its a pretty good chance you are going to get a lot of primes in the sequence as you have excluded the smallest possible factors so any factor much be fairly big. A simpler example is $4p^2-2p+1$. See e.g. https://math.stackexchange.com/questions/439934/quadratic-expression-that-generate-primes (which links to https://www.youtube.com/watch?v=iFuR97YcSLM which you might find interesting ) – Winther Nov 18 '21 at 14:04
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    Thank you, through this formula, we may find unknown prime numbers because this formula gives it a small prime number, which gives you a very large prime number. Perhaps if we try this formula on huge numbers, we will find a very, very large prime number. Example: 2713^3 - 2712^2=19961326153 ،My idea is that maybe if we try on large prime numbers, there is a possibility that we will find prime numbers that are not folded, because the formula through a small prime number finds a large prime number, as the example – Mohammed Nov 18 '21 at 14:11
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    If you try this formula on all numbers less than a billion then under 10% of the numbers will be prime and if you go higher the percentage just keeps dropping. So its not a great way of finding primes. But this kind of idea, using more suitable functions, have been used to find huge primes. You might want to read about Fermat numbers ( https://en.wikipedia.org/wiki/Fermat_number) or Mersenne primes ( https://en.wikipedia.org/wiki/Mersenne_prime ) – Winther Nov 18 '21 at 14:19
  • How is the percentage determined for any formula because I always do formulas about prime numbers, how do I know the percentage of prime numbers that are met? – Mohammed Nov 18 '21 at 14:38
  • By doing it brute force on a computer: check each number and count the number of primes. There is no general method (I think - though one can probably use some probabilistic methods to estimate it very very roughly) since for some simple polynomials like the one mentioned above, $n^2+1$, we don't even know if there are infinite number of primes or not in that sequence. – Winther Nov 18 '21 at 14:49
  • I mean to know how many prime numbers this can be achieved through this formula in a in the first billion of positive integers, I know that the subject of infinity is complex to prove I asked his question only, maybe someone has another explanation – Mohammed Nov 18 '21 at 15:26
  • @Mohammed Since you asked a very similar question , again : $(1)$ We can safely assume that infinite many primes $p$ do the job, but we cannot prove it. $(2)$ There is no easy way to find large examples, in particular if your goal is to find a new record prime. Also, primality proofs only work in special cases or for not too large numbers. – Peter Nov 19 '21 at 14:27
  • $(3)$ To determine the fraction, there is nothing better than brute force. Sieving out small factors of $p$ and $q$ can slightly accelerate the search. – Peter Nov 19 '21 at 14:28
  • Thank you very much. I am not a mathematician. Many things may not be clear to me. I care about numbers and patterns inside them, especially prime numbers.

    When I have an equation like this or another that carries the idea of ​​generating prime numbers between them like Mersenne and others, what is the path that I must go in in order for it to be significant with the mention of two main headings that I even look for.

    I thank you again and the rest of the members, I am new here, but I was glad to help you

     – Mohammed Nov 19 '21 at 15:50

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