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Conjecture: There exists no prime number p > 5 such that both p + 2 and p - 2 are prime numbers.

Formal Proof:

Consider any prime number p greater than 5.

p is an odd prime number, and therefore:

p - 2 and p + 2 are both odd numbers. Every set of three consecutive integers includes a multiple of 3. Since p > 5, we have:

p - 2 > 3 p + 2 > 7 Now, we consider the triplet (p - 2, p, p + 2):

At least one of the numbers in this triplet will be divisible by 3, due to the nature of consecutive integers. A prime number greater than 3 cannot be divisible by 3. Therefore, at least one of the numbers in the triplet (p - 2, p, p + 2) is not prime.

Hence, for any prime number p > 5, it is impossible for both p + 2 and p - 2 to be prime numbers.

Further, there 109,699 prime triplets of the form (p − n, p, p + n) up to 10,000 which cannot be a simple coincidence. The data speaks for itself.enter image description here

Martin.s
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    What's the question? – lulu Oct 03 '23 at 02:21
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    I see no reason your statement about (p-n,p,p+n) cannot be coincidence. Please offer some justification for that. – coffeemath Oct 03 '23 at 02:22
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    There are a lot of small primes, hence a lot of apparent patterns amongst them. Some are real, others not so clear – lulu Oct 03 '23 at 02:25
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    The proposition about the triples $(p - 2, p, p + 2)$ is not a conjecture, it is a theorem (or, more accurately, a common exercise in elementary number theory). It is well known that $(3, 5, 7)$ is the largest such triple, for the reasons you gave. If you're interested in $(p - n, p, p + n)$, you might enjoy reading about Primes in arithmetic progression. – Theo Bendit Oct 03 '23 at 02:32
  • Yes. I apologize for being ill-informed. I will enjoy reading that information. Thank you all for the kind gesture. – Martin.s Oct 03 '23 at 06:06
  • This information is enjoyable for me to read, as expected. – Martin.s Oct 03 '23 at 06:06

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