7.-Conjecture or theorem?
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From: Ludovicus on 4 Aug 2010 07:55 Ribemboim in "The New Book of Prime Numbers Records"(1996) pag 250 says: ".... it is also not known that every even natural number is a difference of two primes (even without requiring them to be consecutive)." I suspect that that is a theorem and easily demonstrable. Ludovicus
From: Gerry on 4 Aug 2010 09:00 On Aug 4, 9:55 pm, Ludovicus <luir...(a)yahoo.com> wrote: > Ribemboim in "The New Book of Prime Numbers Records"(1996) pag 250 > says: > ".... it is also not known that every even natural number is a > difference of two > primes (even without requiring them to be consecutive)." > I suspect that that is a theorem and easily demonstrable. I suspect that you don't know what you're talking about. -- GM
From: Chip Eastham on 4 Aug 2010 09:49 On Aug 4, 7:55 am, Ludovicus <luir...(a)yahoo.com> wrote: > Ribemboim in "The New Book of Prime Numbers Records"(1996) pag 250 > says: > ".... it is also not known that every even natural number is a > difference of two > primes (even without requiring them to be consecutive)." > I suspect that that is a theorem and easily demonstrable. > Ludovicus Ribenboim's editor is lax in places in this book, but this statement is still correct as far as what's "not known." Perhaps you have in mind to consider an arithmetic sequence Pk + E where E is the given even number and P some (odd) number relatively prime to E. Such a sequence will have infinitely many primes, but there is no guarantee any two consecutive entries are both prime, which is what the conjecture above would entail (for some P). regards, chip
From: Ludovicus on 4 Aug 2010 19:38
On 4 ago, 09:49, Chip Eastham <hardm...(a)gmail.com> wrote: > On Aug 4, 7:55 am, Ludovicus <luir...(a)yahoo.com> wrote: > > > Ribemboim in "The New Book of Prime Numbers Records"(1996) pag 250 > > says: > > ".... it is also not known that every even natural number is a > > difference of two > > primes (even without requiring them to be consecutive)." > > I suspect that that is a theorem and easily demonstrable. > > Ludovicus > > Ribenboim's editor is lax in places in this book, but > this statement is still correct as far as what's "not > known." > > Perhaps you have in mind to consider an arithmetic > sequence Pk + E where E is the given even number and > P some (odd) number relatively prime to E. Such a > sequence will have infinitely many primes, but there > is no guarantee any two consecutive entries are both > prime, which is what the conjecture above would entail > (for some P). > > regards, chip Thanks Chip That was exactly what I had in mind. I found, experimentally that the first 30000 even numbers can make true the conjecture with only the primes from 3 to 151. Immediately I think in Dirichlet's theorem. Regards. Luis |