Conjecture on integer sequences
in [Math]
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From: Ludovicus on 8 Jul 2010 11:30 On 8 jul, 11:03, Tim Little <t...(a)little-possums.net> wrote: > On 2010-07-08, Ludovicus <luir...(a)yahoo.com> wrote: > > > New wording of Ludovicus Conjecture: > [...] > > It is supposed that the functions floor or ceiling are not used." > > Heh, a classic application of refining a conjecture by ruling out very > specific inconvenient counterexamples without thinking about the > principles illustrated by those counterexamples. > > - Tim Concerning primes, necessarily, one must ruling out the chopping of of the functions. The conjecture is refered to a property of primes as integers. Otherwise you can manipulate the reals for producing any sequence. For example: Floor[2.0845^(1.221^n)] gives primes from n= 0 to n=10 and then gibberish. Ludovicus
From: jbriggs444 on 8 Jul 2010 12:32 On Jul 8, 11:30 am, Ludovicus <luir...(a)yahoo.com> wrote: > On 8 jul, 11:03, Tim Little <t...(a)little-possums.net> wrote: > > > On 2010-07-08, Ludovicus <luir...(a)yahoo.com> wrote: > > > > New wording of Ludovicus Conjecture: > > [...] > > > It is supposed that the functions floor or ceiling are not used." > > > Heh, a classic application of refining a conjecture by ruling out very > > specific inconvenient counterexamples without thinking about the > > principles illustrated by those counterexamples. > > > - Tim > > Concerning primes, necessarily, one must ruling out the chopping of > of the functions. The conjecture is refered to a property of primes > as integers. Otherwise you can manipulate the reals for producing > any sequence. For example: Floor[2.0845^(1.221^n)] gives primes > from n= 0 to n=10 and then gibberish. > Ludovicus What constitutes an "algorithm" in your book? Do you wish to restrict yourself to real-valued expressions in one positive integer variable using the binary operators for addition, subtraction, multiplication and division together with the unary functions sin, cos, exp and log and the constants 0 and 1 taken in any combination under the restriction that all intermediate results must be real-valued? That's extraordinarily restrictive. Personally, I was thinking along the lines of a Turing machine that takes two integers i and j as input and produces the j'th digit in the decimal expansion of the i'th element in the sequence as output. How do you encode a restriction against using "floor" in such a model?
From: Tim Little on 8 Jul 2010 12:40 On 2010-07-08, Ludovicus <luiroto(a)yahoo.com> wrote: > Concerning primes, necessarily, one must ruling out the chopping of > of the functions. The conjecture is refered to a property of primes > as integers. Otherwise you can manipulate the reals for producing > any sequence. For example: Floor[2.0845^(1.221^n)] gives primes from > n= 0 to n=10 and then gibberish. I would hope that you are aware that in the case of my counterexample, "floor" notation was simply a convenient descriptive shorthand for an integer-only algorithm. If not, you may find it a useful exercise to construct such an algorithm. - Tim
From: OwlHoot on 9 Jul 2010 09:49 On Jul 8, 3:32 pm, Ludovicus <luir...(a)yahoo.com> wrote: > On 7 jul, 11:51, 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 > > New wording of Ludovicus Conjecture: > "If by the uniform application of an algorithm, an infinite integer > sequence Un results, such that LogLog(Un) is of the order of Log(n) > or less, and it contains five or more primes of certain type, > (Not counting the numbers given in the definition of the algorithm) > then it will contains infinitely many primes of that type. > It is supposed that the functions floor or ceiling are not used." > > Examples: In twin primes , Fibonacci,Mersenne,and polynomial > sequences the two conditions are fulfiled. > But not in Fermat's sequence because LogLog(Fn) = 2^n + Log2. > Nor in Un = 10^n + 1 because only 11 and 101 are known as primes. > Ludovicus I think this is a counterexample: Choose your five favourite primes p1, p2, p5, or just take 2, 3, 5, 7, 11. Then the following sequence contains exactly 5 primes, and satisfies your growth conditions up to a constant: { p1^1 . p2^0 . p3^0 . p4^0 . p5^0, p1^0 . p2^1 . p3^0 . p4^0 . p5^0, ::: p1^0 . p2^0 . p3^0 . p4^0 . p5^1, p1^2 . p2^0 . p3^0 . p4^0 . p5^0, p1^0 . p2^2 . p3^0 . p4^0 . p5^0, ::: p1^0 . p2^0 . p3^0 . p4^0 . p5^2, ::::: p1^n . p2^0 . p3^0 . p4^0 . p5^0, p1^0 . p2^n . p3^0 . p4^0 . p5^0, ::: p1^0 . p2^0 . p3^0 . p4^0 . p5^n, ::::::::::: and no floors or ceilings etc in sight ;-) (or primes after the first five) Cheers John Ramsden
From: OwlHoot on 9 Jul 2010 10:08
On Jul 9, 2:49 pm, OwlHoot <ravensd...(a)googlemail.com> wrote: > On Jul 8, 3:32 pm, Ludovicus <luir...(a)yahoo.com> wrote: > > > > > On 7 jul, 11:51, 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 > > > New wording of Ludovicus Conjecture: > > "If by the uniform application of an algorithm, an infinite integer > > sequence Un results, such that LogLog(Un) is of the order of Log(n) > > or less, and it contains five or more primes of certain type, > > (Not counting the numbers given in the definition of the algorithm) > > then it will contains infinitely many primes of that type. > > It is supposed that the functions floor or ceiling are not used." > > > Examples: In twin primes , Fibonacci,Mersenne,and polynomial > > sequences the two conditions are fulfiled. > > But not in Fermat's sequence because LogLog(Fn) = 2^n + Log2. > > Nor in Un = 10^n + 1 because only 11 and 101 are known as primes. > > Ludovicus > > I think this is a counterexample: > > Choose your five favourite primes p1, p2, p5, or just take > 2, 3, 5, 7, 11. > > Then the following sequence contains exactly 5 primes, and > satisfies your growth conditions up to a constant: > > { p1^1 . p2^0 . p3^0 . p4^0 . p5^0, > > p1^0 . p2^1 . p3^0 . p4^0 . p5^0, > > ::: > > p1^0 . p2^0 . p3^0 . p4^0 . p5^1, > > p1^2 . p2^0 . p3^0 . p4^0 . p5^0, > > p1^0 . p2^2 . p3^0 . p4^0 . p5^0, > > ::: > > p1^0 . p2^0 . p3^0 . p4^0 . p5^2, > > ::::: > > p1^n . p2^0 . p3^0 . p4^0 . p5^0, > > p1^0 . p2^n . p3^0 . p4^0 . p5^0, > > ::: > > p1^0 . p2^0 . p3^0 . p4^0 . p5^n, > > ::::::::::: > > and no floors or ceilings etc in sight ;-) > > (or primes after the first five) > > Cheers > > John Ramsden Actually, one might just as well express this sequence more compactly as: {p1, p2, .. , p5, p1^2, p2^2, .. p5^2, p1^3, ... } JR |