Prime numbers and primality testing is a Restricted Group with 1137 members. Yahoo Groups 1993/2011 puzzle Kevin Acres Message 1 of 15 , Jan 8, 2011 ----------------------- I'm not sure how obvious this is but: Can anyone tell me what the relation is between 1993 and 2011 as regards a certain type of prime. Further clues are the year of the first human blood transfusion and the year that Henry IV of England was crowned. Best Regards, Kevin. mikeoakes2 Jan 8, 2011 ----------------------- --- In primenumbers@yahoogroups.com, Kevin Acres wrote: > > I'm not sure how obvious this is but: > > Can anyone tell me what the relation is between 1993 and 2011 as > regards a certain type of prime. > > Further clues are the year of the first human blood transfusion and > the year that Henry IV of England was crowned. A mindless search of OEIS shows that there is no (relevant) sequence containing the 2 primes 1667 and 1399 - so your puzzle clearly needs more thought than that:-) Mike Kevin Acres Jan 8, 2011 ----------------------- I just checked OEIS and it doesn't give this game away, nor really offer any help. Although I have no historical data for the appropriate year, I should maybe add that 257 would be a member of the sequence, along with 1339, 1993 and 2011, if such existed on OEIS. 1 new number, two new clues :-) Best Regards, Kevin. At 03:35 AM 9/01/2011, mikeoakes2 wrote: Show message history Kevin Acres Jan 8, 2011 ----------------------- Sorry, typo. 257, 1339, 1667 and 2011 would be some of members of the list. Below I wrote 1993 in place of 1667. At 07:09 AM 9/01/2011, Kevin Acres wrote: Show message history Kevin Acres Jan 8, 2011 ----------------------- I really shouldn't try and type anything meaningful this time of the morning. Third try for 4 consecutive numbers in the list: 257, 1399, 1667, 2011 Apologies for the waste of bandwidth. At 07:25 AM 9/01/2011, Kevin Acres wrote: Show message history mikeoakes2 Jan 8, 2011 ----------------------- --- In primenumbers@yahoogroups.com, Kevin Acres wrote: > > I just checked OEIS and it doesn't give this game away, nor really > offer any help. > > Although I have no historical data for the appropriate year, I should > maybe add that 257 would be a member of the sequence, along with > 1339, 1993 and 2011, if such existed on OEIS. How about this important event: http://www.newadvent.org/fathers/3818.htm Mike Kevin Acres Jan 8, 2011 ----------------------- At 08:44 AM 9/01/2011, mikeoakes2 wrote: >--- In primenumbers@yahoogroups.com, Kevin Acres wrote: > > > > I just checked OEIS and it doesn't give this game away, nor really > > offer any help. > > > > Although I have no historical data for the appropriate year, I should > > maybe add that 257 would be a member of the sequence, along with > > 1339, 1993 and 2011, if such existed on OEIS. > >How about this important event: >http://www.newadvent.org/fathers/3818.htm > >Mike Well spotted Mike. 257 really is 2 clues in 1 especially if I mention that 5 also appears in the list -> 5, 257, 1399. 1667, 2011. These primes all share the same relationship with 1993. Which is all probably quite enough to give the game away. Kevin. djbroadhurst Jan 8, 2011 ----------------------- --- In primenumbers@yahoogroups.com, Kevin Acres wrote: > 257 really is 2 clues in 1 especially if I mention that 5 also > appears in the list -> 5, 257, 1399. 1667, 2011. > These primes all share the same relationship with 1993. I am utterly defeated by this puzzle. Perhaps it is some of retaliation for > Australia lost inns & 25 runs v West Indies Perth 30 Jan 1993 as posted by Andy :-? David Kevin Acres Jan 8, 2011 ----------------------- Hello David, It's not in retaliation, it's just the first puzzle using the prime 2011 that came to mind given the reason behind my latest x=a*x+b search. At 03:12 PM 9/01/2011, djbroadhurst wrote: >--- In primenumbers@yahoogroups.com, >Kevin Acres wrote: > > > 257 really is 2 clues in 1 especially if I mention that 5 also > > appears in the list -> 5, 257, 1399. 1667, 2011. > > These primes all share the same relationship with 1993. > >I am utterly defeated by this puzzle. > >Perhaps it is some of retaliation for > > Australia lost inns & 25 runs v West Indies Perth 30 Jan 1993 >as posted by Andy :-? > >David I'm not sure what other hints can I give without totally giving the game away. You can be sure that you don't need to think outside the square, as the saying goes. The entire sequence of these primes, below 10^8, is 2, 5, 257, 1399, 1667 and 2011. I am also not aware of any others > 10^8. This sparseness is both a hint and a feature shared by the generic family. Of course, once you see the solution the duality of clues from 257 will immediately become clear. Those who absolutely give up may quickly locate a major clue to arriving at the correct solution by googling the name of this group followed by 1909 :-) Best Regards, Kevin (with obviously way too much time on his hands). djbroadhurst Jan 9, 2011 ----------------------- --- In primenumbers@yahoogroups.com, Kevin Acres wrote: > 257 really is 2 clues in 1 especially if I mention that 5 also > appears in the list -> 5, 257, 1399. 1667, 2011. > These primes all share the same relationship with 1993. ... > You can be sure that you don't need to think outside the square, > as the saying goes. ... > Those who absolutely give up may quickly locate a major clue > to arriving at the correct solution by googling the name of > this group followed by 1909 :-) The penny finally dropped :-) Another hint: a quotient too big for the margin. David djbroadhurst Jan 9, 2011 ----------------------- --- In primenumbers@yahoogroups.com, "djbroadhurst" wrote: > The penny finally dropped :-) > Another hint: a quotient too big for the margin. Puzzle [1993/2001/32933]: Find the next prime in this minimally increasing sequence of primes: 1993, 2011, 32933 ... David djbroadhurst Jan 9, 2011 ----------------------- --- In primenumbers@yahoogroups.com, "djbroadhurst" wrote: > Another hint: a quotient too big for the margin. Kevin has asked me to post my solution. The key to his puzzle was http://en.wikipedia.org/wiki/Fermat_quotient With base a = 1993, the Fermat quotient (a^(p-1) - 1)/p is divisible by p for p = 2, 5, 257, 1399, 1667, 2011 and for no other prime p < 10^10. In other words, 2011 is the largest known Wieferich prime in base 1993. Another puzzle is to find a sequence of increasing primes such that p[n+1] is the smallest Wieferich prime in base p[n] for which p[n+1] > p[n]. Let's call this a "generalized Wieferich sequence". For example, the primes 1153, 1747, 1993, 2011, 32933, 16365127 form a generalized Wieferich sequence of length 6, with 16365127 being Kevin's correct solution to my "1993/2011/32933 puzzle". Puzzle 7: Find a generalized Wieferich sequence of length > 6. Comments: This may be solved using primes less than 10^9. So far, I have not found a generalized Wieferich sequence of length > 7. David djbroadhurst Jan 9, 2011 ----------------------- --- In primenumbers@yahoogroups.com, "djbroadhurst" wrote: > Another puzzle is to find a sequence of increasing primes > such that p[n+1] is the smallest Wieferich prime in base p[n] > for which p[n+1] > p[n]. > > Let's call this a "generalized Wieferich sequence". > > For example, the primes > 1153, 1747, 1993, 2011, 32933, 16365127 > form a generalized Wieferich sequence of length 6, with > 16365127 being Kevin's correct solution to my > "1993/2011/32933 puzzle". > > Puzzle 7: Find a generalized Wieferich sequence of length > 6. My interest in generalized Wieferich primes comes from the great achievement of Preda Mihailescu http://en.wikipedia.org/wiki/File:450px-Preda_Mihailescu_vor_Tafel.png one of the authors of OpenPFGW, who proved the Catalan Conjecture: http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf Note that the reviewer says that Preda was "a mathematician practically unknown to the experts in this area" who "turned up with a complete proof of the conjecture." Moreover, "his proof has very little to do with computation, making instead use of deep theoretical results, notably from the theory of cyclotomic fields. Mihailescu, born in 1955 in Romania, received his mathematical education at the ETH Zurich. He has worked in the machine and finance industry but is now doing research in Germany at the University of Paderborn. The present article describes briefly the landmarks in the history of the work on Catalan's problem and outlines Mihailescu's brilliant solution." So next time that someone tells you that you are "unknown to the experts", remember Preda :-) David Makoto Kamada Jan 9, 2011 ----------------------- > Puzzle 7: Find a generalized Wieferich sequence of length> 6. 197, 653, 1381, 1777, 6211, 39041, 144449603 Makoto Kamada djbroadhurst Message 15 of 15 , Jan 9, 2011 ----------------------- --- In primenumbers@yahoogroups.com, Makoto Kamada wrote: >> Puzzle 7: Find a generalized Wieferich sequence of length > 6. > 197, 653, 1381, 1777, 6211, 39041, 144449603 Congratulations to Makoto for a valid solution, with the last 2 primes recorded in http://www.cecm.sfu.ca/~mjm/WieferichBarker/Data/q5p9.txt In addition to Makoto's solution, I found the sequence 19739, 61729, 445631, 508009, 728437, 1051139, 5176948723 Now comes a harder puzzle, which I have not solved :-( > Another puzzle is to find a sequence of increasing primes > such that p[n+1] is the smallest Wieferich prime in base p[n] > for which p[n+1] > p[n]. > Let's call this a "generalized Wieferich sequence". Puzzle 8: Find a generalized Wieferich sequence of length > 7. David