According to my information the evidence, that there are infinity primes is reasoned through the Euclid-Numbers which always are prime: \(P_{1} \times P_{2} \times ... \times P_{n} = N⇒[N+ 1] ∈ Primes\). Since you always have the opportunity, to multiply all the previous primes and add 1, there must be infinity primes.
Now I've heard 2 interesting facts:
It makes good sense, that not all Euclid-Numbers necessarily have to be prime. It's proven, that the Euclid-Numbers aren't divisible by any number between and (because there will always be a remainder of 1), but what about ? There still are other primes, that could be factors of . The first Euclid-Number, that isn't a prime is 30031 = 59 × 509 (it's the 6th Euclid-Number). Here the trick doesn't work anymore, because took primes between and . So why is this still a prove? Of course it seems very unlikely, that there is an end for the primes, but is it still a prove?
The reason, why this Euclid-Numbers-Trick works is, because the formula makes sure, that the most often used (least) primes can't be a factor of and this makes the probability to get a composite number much worse. But actually we can do the same trick with instead of . if we do , there will be a remainder of when we divide it by any integer between and . So But why isn't this a prove, that there are infinity twinprimes, though it uses the principle of the proof, that there are infinity primes (or am I a professor now 🙂)?
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Top NewestAs shown in the first part, if there is a prime factor to an Euclid number (never knew they were called like that) it is surely bigger than all the primes taken into account (in calculating the Euclid number). If they are no prime factors to the Euclid number then it is a prime itself which completes the proof that there will be always a prime greater than all the primes we discover. But for the second part N + 1 and N - 1 may not be primes themselves, just having prime factors greater than the primes used to calculate N. There is no guarantee that as we approach bigger numbers, N + 1 and N - 1 will be prime. ( I don’t think u became a professor :) )
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Thank you! I've finally found the error in reasoning 😁 (I'm so stupid xD).
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Lol no, we all get excited sometimes, I used to feel I could square a circle before, although knowing that it was mathematically proven that it was impossible xD
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