dave (npub1tsg…htk4) check out this CHAT GPT proof for the twin prime conjecture for P>5(mod6)
There are infinitely many prime numbers. This is a well-known result proven by the ancient Greek mathematician Euclid around 300 BCE. Euclid's proof is based on a technique called "proof by contradiction."
Assume that there are finitely many prime numbers, and let's denote them as p1, p2, p3, ..., pn. Now, consider the number N = p1 * p2 * p3 * ... * pn + 1. This number N is greater than any of the primes we initially listed.
Now, N can either be a prime number itself or composite. If N is prime, then it contradicts our assumption that we had listed all prime numbers up to pn. If N is composite, it must have prime factors. However, none of the primes p1, p2, ..., pn can be a factor of N because dividing N by any of them leaves a remainder of 1. This means that N has a prime factor that is not on our initial list, contradicting our assumption that we had all primes.
Since assuming a finite number of primes leads to a contradiction, we conclude that there must be infinitely many prime numbers.
Now let's discuss why there are infinitely many prime numbers congruent to 1 (mod 6) and infinitely many prime numbers congruent to 5 (mod 6).
Suppose we consider the product of prime numbers that are congruent to 1 (mod 6). For example, we can take the primes 7, 13, 19, 31, and so on. The product of these primes will also be congruent to 1 (mod 6). This is because multiplying numbers congruent to 1 (mod 6) will always result in a number congruent to 1 (mod 6).
Similarly, if we consider the product of prime numbers that are congruent to 5 (mod 6), such as 5, 11, 17, 23, and so on, the product will also be congruent to 5 (mod 6). This is because multiplying numbers congruent to 5 (mod 6) will always result in a number congruent to 5 (mod 6).
Since there are infinitely many prime numbers, and we can always find more primes congruent to 1 (mod 6) and primes congruent to 5 (mod 6), it follows that there are infinitely many such products congruent to 1 (mod 6) and infinitely many products congruent to 5 (mod 6).
Therefore, both the product of primes congruent to 1 (mod 6) and the product of primes congruent to 5 (mod 6) are infinite.
dula / DULA
npub1ct…wucne
2023-06-16 00:25:13
Author Public Key
npub1ctagm8mys2he926jqxcfph4vz3kjqzmxxch5ysstxqxtvl0uuz4qgwucnePublished at
2023-06-16 00:25:13Event JSON
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"content": "#[0] check out this CHAT GPT proof for the twin prime conjecture for P\u003e5(mod6)\n\nThere are infinitely many prime numbers. This is a well-known result proven by the ancient Greek mathematician Euclid around 300 BCE. Euclid's proof is based on a technique called \"proof by contradiction.\"\n\nAssume that there are finitely many prime numbers, and let's denote them as p1, p2, p3, ..., pn. Now, consider the number N = p1 * p2 * p3 * ... * pn + 1. This number N is greater than any of the primes we initially listed.\n\nNow, N can either be a prime number itself or composite. If N is prime, then it contradicts our assumption that we had listed all prime numbers up to pn. If N is composite, it must have prime factors. However, none of the primes p1, p2, ..., pn can be a factor of N because dividing N by any of them leaves a remainder of 1. This means that N has a prime factor that is not on our initial list, contradicting our assumption that we had all primes.\n\nSince assuming a finite number of primes leads to a contradiction, we conclude that there must be infinitely many prime numbers.\n\nNow let's discuss why there are infinitely many prime numbers congruent to 1 (mod 6) and infinitely many prime numbers congruent to 5 (mod 6).\n\nSuppose we consider the product of prime numbers that are congruent to 1 (mod 6). For example, we can take the primes 7, 13, 19, 31, and so on. The product of these primes will also be congruent to 1 (mod 6). This is because multiplying numbers congruent to 1 (mod 6) will always result in a number congruent to 1 (mod 6).\n\nSimilarly, if we consider the product of prime numbers that are congruent to 5 (mod 6), such as 5, 11, 17, 23, and so on, the product will also be congruent to 5 (mod 6). This is because multiplying numbers congruent to 5 (mod 6) will always result in a number congruent to 5 (mod 6).\n\nSince there are infinitely many prime numbers, and we can always find more primes congruent to 1 (mod 6) and primes congruent to 5 (mod 6), it follows that there are infinitely many such products congruent to 1 (mod 6) and infinitely many products congruent to 5 (mod 6).\n\nTherefore, both the product of primes congruent to 1 (mod 6) and the product of primes congruent to 5 (mod 6) are infinite.",
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