Conjecture on integer sequences
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From: Ludovicus on 7 Jul 2010 11:51 Conjecture: If a sequence Un is such that LogLog(Un) is of the order Log(n) or less and it contains five or more primes, then the sequence will contains infinitely many primes. (No counting in the five, the numbers used to initiate the algorithm.) This conjecture will comprise many of the unsolved prime conjectures: Twin primes, Fibonacci primes, Polynomial primes, Mersenne primes, etc But not Fermat primes. Ludovicus
From: Ross on 7 Jul 2010 12:38 On Jul 7, 8:51 am, Ludovicus <luir...(a)yahoo.com> wrote: > Conjecture: > If a sequence Un is such that LogLog(Un) is of the order Log(n) or > less > and it contains five or more primes, then the sequence will contains > infinitely many primes. (No counting in the five, the numbers used to > initiate the algorithm.) > This conjecture will comprise many of the unsolved prime conjectures: > Twin primes, Fibonacci primes, Polynomial primes, Mersenne primes, etc > But not Fermat primes. > Ludovicus You need to define what sequences you are considering. How about: Un=2,3,5,7,11 for n=1 to 5 Un=10^n for n>=6?
From: Ludovicus on 7 Jul 2010 13:08 On 7 jul, 12:38, Ross <rmill...(a)pacbell.net> wrote: > On Jul 7, 8:51 am, Ludovicus <luir...(a)yahoo.com> wrote: > > > Conjecture: > > If a sequence Un is such that LogLog(Un) is of the order Log(n) or > > less > > and it contains five or more primes, then the sequence will contains > > infinitely many primes. (No counting in the five, the numbers used to > > initiate the algorithm.) > > This conjecture will comprise many of the unsolved prime conjectures: > > Twin primes, Fibonacci primes, Polynomial primes, Mersenne primes, etc > > But not Fermat primes. > > Ludovicus > > You need to define what sequences you are considering. How about: > Un=2,3,5,7,11 for n=1 to 5 > Un=10^n for n>=6? Yes . I forget to stablish that the sequence must be infinite and produced by the uniform application of a given algorithm. Your example do not conform with the conditions because your five numbers are precisely, the given for initiate the algorithm. Ludovicus
From: OwlHoot on 7 Jul 2010 14:04 On Jul 7, 4:51 pm, Ludovicus <luir...(a)yahoo.com> wrote: > Conjecture: > If a sequence Un is such that LogLog(Un) is of the order Log(n) or > less > and it contains five or more primes, then the sequence will contains > infinitely many primes. (No counting in the five, the numbers used to > initiate the algorithm.) > This conjecture will comprise many of the unsolved prime conjectures: > Twin primes, Fibonacci primes, Polynomial primes, Mersenne primes, etc > But not Fermat primes. > Ludovicus Couldn't you just define the sequence to be the first 10 primes, and the primes plus 1 thereafter? (assuming the infinite sequence of primes has the property) Cheers John Ramsden
From: Ludovicus on 7 Jul 2010 14:37
On 7 jul, 14:04, OwlHoot <ravensd...(a)googlemail.com> wrote: > On Jul 7, 4:51 pm, Ludovicus <luir...(a)yahoo.com> wrote: > > > Conjecture: > > If a sequence Un is such that LogLog(Un) is of the order Log(n) or > > less > > and it contains five or more primes, then the sequence will contains > > infinitely many primes. (No counting in the five, the numbers used to > > initiate the algorithm.) > > This conjecture will comprise many of the unsolved prime conjectures: > > Twin primes, Fibonacci primes, Polynomial primes, Mersenne primes, etc > > But not Fermat primes. > > Ludovicus > > Couldn't you just define the sequence to be the first 10 primes, > and the primes plus 1 thereafter? (assuming the infinite sequence > of primes has the property) > > Cheers > > John Ramsden I choose 5 as the needed number of prime based in the five Fermat primes because LogLog(Fn) = 2^n Log2 > Log(n). That is Fermat's do not guarantee infinitely many primes, but fulfils the first condition. In the case Un = n^2 + n + 13, five primes results in n = 0, 2, 5, 9, 14 Ludovicus |