Anyone can participate with just a computer connection and willingness to give up a few computational resources. The 48 TH Mersenne Prime is the latest discovery posted on the site, but this can change at any moment. In considering the infinitude of Mersenne Primes it would be interesting to see the emergence of an elegant solution.
It is neat to ponder such possibilities. Could it come from a proof by contradiction? Maybe it will be trivially evident or a consequence of some other discovery. Can you link it to another conjecture or theorem? Whatever the case, problems like these fuel the fire of ambition and spark the imagination of mathematicians, amateur and professional alike. Your email address will not be published.
We may rigorously prove formulas 3. Let T e be the set of all S-primes. Next we prove that the cardinality of the set T e is infinite by existing theories of those structures,. Based on the recursive algorithm, formula 3. We delete non S-primes or non S-primes together with a S-prime. The recursive sieve 4. So that we only need to determine the number of all S-primes T e. If we do so successfully, then the parity obstruction, a ghost in house of primes, has been automatically evaporated.
With the recursive sieve 4. Let A i be the set of all S-primes x less than p i. From the recursive formula 3.
Now we intercept the initial segment from the left set L i , which is the union of the set A i of S-primes and the set T i of least nonnegative representatives. Then we obtain a new recursive formula. Formula 4. The initial segment is a well chosen notation.
We shall consider some properties of the initial segment, and reveal some structures of the sequence of the initial segments to determine the set of all S-primes and its cardinality. Let A i be the number of S-primes less than p i. From formula 3. Thus the set of all S-primes is limit computable and is an infinite set. For the convenience of the reader, we quote a definition of the set theoretic limit of a sequence of sets [7]. We know that the sequence of left sets L i is descending.
According to the definition of the set theoretic limit of a sequence of sets, we obtain that the sequence of left sets L i converges to the set T e. The sequence of subsets A i.
We obtain that the sequence of subsets A i. Thus the sequence of the initial segments T i. According to set theory, we have proved that both sequences of sets T i. Next we prove that according to order topology both sequences of sets T i.
Let X be a set with a linear order relation; assume X has more one element. Let B be the collection of all sets of the following types:.
According to the definition there is no order topology on the empty set or sets with a single element. The recursively sifting process, formula 4.
We further consider the structures of sets X 1 and X 2 using the recursively sifting process 4. The set X 1 has no repeated term. So if you wanted something to be widely known, you told him. Add a comment. This work is available in more detail in the appendix of my recent paper The supplementary materials contain example MS Excel spreadsheets illustrating my approach. Grenville Croll Grenville Croll 69 4 4 bronze badges. Sign up or log in Sign up using Google.
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Featured on Meta. However, as with — 1, we know not all Mersenne numbers of form 2p — 1 are primes. All of this evidence makes it reasonable to conjecture that there exist an infinite number of Mersenne primes. First we will provide additional evidence indicating an infinite number of Mersenne primes.
Then we will provide the proof. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary.
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