3 and 7 are primes, and (3+7)%2 is 5 which is a prime too. But 3 and 7 are not consecutive primes, because there is a prime between 3 and 7, namely 5. So if p < q and p is prime and q is prime and r= (p+q)%2 is prime too, then p < r and r < q, and so p and q are not consecutive primes. QED. There is no need for induction.
>________________________________ > Fra: Roger Hui <rogerhui.can...@gmail.com> >Til: Programming forum <programm...@jsoftware.com> >Sendt: 4:01 mandag den 13. maj 2013 >Emne: Re: [Jprogramming] Testing consecutive pairs of primes > > >Henry has already argued that if p and q are consecutive primes then >(p+q)%2 can not be prime. I just want to say that reasoning of the sort: > >While it might be possible for the larger primes, I'm thinking not - just >by induction. > > >On Sun, May 12, 2013 at 3:07 AM, Alan Stebbens <a...@stebbens.org> wrote: > >> ProgrammingPraxis (at http://programmingpraxis.com/2013/05/10/mindcipher) >> offered a problem asking, given p, q as two consecutive pairs of primes, if >> (p+2)%2 could be prime. >> >> Since both p & q (> 2) are prime, their sum is an even number and not >> prime, but could the half of their sum be a prime? >> >> I'm not much of a mathematician, but I figured I could brute-force an >> approximation with J. >> >> The gist below is my experiment showing that the answer is no, for the >> consecutive pairs of primes in the set of the first million primes. While >> it might be possible for the larger primes, I'm thinking not - just by >> induction. >> >> Probably some of you could show a proof, but it was more fun for me to >> cobble up this in J, and also demonstrate J to the non-J audience at >> Programming Praxis (which has been mostly scheme, Haskell, python, ruby). >> >> https://gist.github.com/aks/5563008 >> >> Here's my experiment: >> >> i. 20 >> 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 >> >> NB. generate the first 20 primes >> p: i. 20 >> 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 >> >> NB. box up consecutive pairs of those primes >> (2 <\ ]) p: i. 20 >> >> ┌───┬───┬───┬────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐ >> │2 3│3 5│5 7│7 11│11 13│13 17│17 19│19 23│23 29│29 31│31 37│37 41│41 43│43 >> 47│47 53│53 59│59 61│61 67│67 71│ >> >> └───┴───┴───┴────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘ >> >> NB. sum up each pair of primes >> +/ each (2 <\ ])p: i. 20 >> ┌─┬─┬──┬──┬──┬──┬──┬──┬──┬──┬──┬──┬──┬──┬───┬───┬───┬───┬───┐ >> │5│8│12│18│24│30│36│42│52│60│68│78│84│90│100│112│120│128│138│ >> └─┴─┴──┴──┴──┴──┴──┴──┴──┴──┴──┴──┴──┴──┴───┴───┴───┴───┴───┘ >> >> NB. divide each sum by 2 >> 2 %~ each +/ each (2 <\ ])p: i. >> 20 >> >> ┌───┬─┬─┬─┬──┬──┬──┬──┬──┬──┬──┬──┬──┬──┬──┬──┬──┬──┬──┐ >> │2.5│4│6│9│12│15│18│21│26│30│34│39│42│45│50│56│60│64│69│ >> └───┴─┴─┴─┴──┴──┴──┴──┴──┴──┴──┴──┴──┴──┴──┴──┴──┴──┴──┘ >> >> NB. now, test each of those results for being prime. 1 p: y -- tests y >> for being prime >> >> 1&p: each 2 %~ each +/ each (2 <\ ])p: i. >> 20 >> >> ┌─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┬─┐ >> │0│0│0│0│0│0│0│0│0│0│0│0│0│0│0│0│0│0│0│ >> └─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┴─┘ >> >> NB. open the boxed results, so we can add them up >> >1&p: each 2 %~ each +/ each (2 <\ ])p: i. 20 >> 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 >> >> NB. sum/reduce the vector of booleans. If there's a prime, the sum will >> be > 0 >> +/>1&p: each 2 %~ each +/ each (2 <\ ])p: i. 20 >> 0 >> >> NB. ok. No primes. Let's keep checking for larger groups >> >> +/>1&p: each 2 %~ each +/ each (2 <\ ])p: i. 1000 >> 0 >> +/>1&p: each 2 %~ each +/ each (2 <\ ])p: i. 10000 >> 0 >> +/>1&p: each 2 %~ each +/ each (2 <\ ])p: i. 100000 >> 0 >> >> NB. the previous output took a few seconds. The next will take a few >> minutes >> >> +/>1&p: each 2 %~ each +/ each (2 <\ ])p: i. 1000000 >> 0 >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm >---------------------------------------------------------------------- >For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
