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galoisjr
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The prime numbers are the multiplicative building blocks of the integers. Even so, their distribution escapes all methods of rationalization. As with building a pyramid, the primes are most densely distributed near zero, the point of origin, and as we move towards larger numbers the primes are less dense. However, counterintuitive to this notion, there do exist arbitrarily large consecutive prime numbers which are separated by an interval of only 2. The Twin Prime Conjecture asserts that these pairs form an infinite set.
The proof of this conjecture has eluded number theorists for quite sometime. Being only in my junior year of my undergraduate education, and never having taken a course in number theory, I do not think that I have the tools to rigorously proof the conjecture. But I do think that I have found a path towards a constructive proof.
I sort of stumbled into this while looking at a list of factorials, but I realized that for quite a few integer j, a ball of radius 1 about j! contains either two primes, or a prime and a square. After toying with this notion for awhile, I made a conjecture of my own which I am fairly certain of:
For each n in the integers there exists some k in the integers such that (k*n!-1, k*n!+1) is a twin prime pair.
I have no idea where to begin a proof of the validity or invalidity of this statement. And this statement alone is not sufficient proof of the Twin Prime Conjecture since there would be some repetition of the twin prime pairs as n increases and we cycle back through the integers to find a suitable k. However, if the statement is true, then finding a bound on k will be sufficient proof of the TPC.
Any comments, skepticism, ideas, etc. are highly encouraged.
Thanks in advance
The proof of this conjecture has eluded number theorists for quite sometime. Being only in my junior year of my undergraduate education, and never having taken a course in number theory, I do not think that I have the tools to rigorously proof the conjecture. But I do think that I have found a path towards a constructive proof.
I sort of stumbled into this while looking at a list of factorials, but I realized that for quite a few integer j, a ball of radius 1 about j! contains either two primes, or a prime and a square. After toying with this notion for awhile, I made a conjecture of my own which I am fairly certain of:
For each n in the integers there exists some k in the integers such that (k*n!-1, k*n!+1) is a twin prime pair.
I have no idea where to begin a proof of the validity or invalidity of this statement. And this statement alone is not sufficient proof of the Twin Prime Conjecture since there would be some repetition of the twin prime pairs as n increases and we cycle back through the integers to find a suitable k. However, if the statement is true, then finding a bound on k will be sufficient proof of the TPC.
Any comments, skepticism, ideas, etc. are highly encouraged.
Thanks in advance
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