Problems & Puzzles:
Puzzles
Puzzle 121. THE LEGEND
OF YANG HUI
I have chosen as the last puzzle of the
current millennium, one issue from a chapter of the book in preparation of my friend Luis
Rodríguez  entitled "The World of Prime Numbers" 
with his kindly and generous permission.
The chosen puzzle deals with an ancient
puzzle supposedly posed by the Emperor Sung to the mathematician Yang Hui.
In Rodríguez's own words:
"...After a laborious work , Yang Hui
could finish in the year 1275 his book 'Hsu Ku Chai Chi Suan Fa' (Continuation
of Ancient Mathematical Methods for Elucidating the Strange Properties of
Numbers). Among other matters this book is about magic squares. He gives the instructions for
construct magic squares of order 3, 4 and 5.
The day he presented the book to old Emperor Sung an enthusiast of
mathematics Yang Hui said: Majesty, here is my gift : a magic square of numbers in arithmetic
progression such that if you add one to each cell results another of prime numbers only.
1668 
198 
1248 

1669 
199 
1249 
618 
1038 
1458 
+1> 
619 
1039 
1459 
828 
1878 
408 

829 
1879 
409 
(Constant difference = 210)
Enchanted the Emperor rewarded him with a splendid steed and said to him afterwards: 'But now, I observe that if one is
subtracted to each cell, not all of they results primes.' (*)
If you bring me a magic square of numbers in arithmetic progression where there results primes either we add or
subtract one, I shall give you a precious villa in the country
of your dreams : Kueilin, at the border of Li river.
Mounting swiftly on his horse, Yang Hui was lost in the distance.... Still the Emperor is waiting for the
return of Yang Hui and his super twinmagic square...".
***
Up to here we follow the Rodríguez's
literary thread.
As you can see, in order to get the asked magic
square we need to discover at least nine
Twin primes in Arithmetic Progression.
As a matter of fact here we are talking about of a double arithmetic
progression, both with the same path (distance) and differing
member to member by two (2). A kind of arithmetic DNA structure...
Question 1: Get one/the
least (**) solution to the Emperor Sun's
request.
Question 2: Get K
Twin primes in A.P. for K>9
(***)
________
Note (*): As a matter of fact only 1667,
197, 617, 827 & 1877 are primes.
Note (**): 'the least' means
here the solution with the minimal largest prime of both progressions.
Note (***): Probably you may think that 9 Twin
primes in A.P. are too hard to get. As an optimistic hint let me tell you
that I have discovered two examples of 8 Twin primes in A.P., both
examples with the largest prime less than 10^8. I can not assure that the
asked 9 twin primes in A.P. are too close to the discovered 8 ones, but it
could happen...
Note about the drawing: This is a
collaboration of Juan Sabastián, Luis's son.
Solution
Well, the winner of the ville KueiLin
is Paul Jobling, and BTW he is the winner since last May. According
to him (30/12/2000) "...I made the search in May of this year, for no reason other
than to look for a long AP of twin primes as I had not seen anything
published on them. I sent my result to the primel mailing list and to the
Number Theory mailing list (nmbrthry)...."
All his solutions are for K = 10 being the
smaller (130864+i*66391)*11#+180+1,
for i=0 to 9... where the common difference is 153363210 (=66391*11#)
and provide the following solution to the Magic square asked by
the Emperor Sung:
1375838489

609022439

762385649

1375838490

609022440

762385650

1375838491

609022441

762385651

302296019

915748859

1529201699

302296020

915748860

1529201700

302296021

915748861

1529201701

1069112069

1222475279

455659229

1069112070

1222475280

455659230

1069112071

1222475281

455659231

green = smallest prime; red
= largest prime
Paul adds "No doubt somebody
will find the smallest example of an AP of 9 twin primes, it will be
smaller than the square that I sent to you"
Regarding the method he employed Paul
wrote:
"We
are looking for an AP of twin primes: a +1, a+b +1, a+2b +1, ...,
a+(n1)b +1. For an AP of twin primes of length n, it is obvious that b
must be a multiple of n#, the product of the primes <=n. I therefore
performed the following search:
For p= the primes 11, 13, 17: For each possible
x <= p#, Obtain the 1 million numbers of the form x + i.p#, 0 <= i
< 1000000 Discard those i's where x+i.p#+1 or x+i.p#1 is composite.
This will leave those i's where x+i.p# + 1 is a pair of twin primes.
Search these i's for an arithmetic progression
of length >= 8. The testing time was very short for p=11 and 13
(hours); for p=17 the search time was approximately 40 days on a 233Mhz
PII."
***
Current status (30/12/2000)
Regarding the Magic square asked by
the Emperor, Paul has gotten several solutions, but his smaller one
probably is not the earliest.
Regarding The
Least Sequences of K Twin primes in A.P. the situation is as
follows:
K

D

First
& lower prime of the least sequence

Last
& upper prime of the sequence 
Auhor 
2

2

3

7 

3

6

5

19 

4

12

5

43 

5

180

101

823 
C.R.,
12/2000 
6

420

41

2143 
Luis
Rodríguez, 12/2000 
7

439320

22367

2658289 
C.R.,
12/2000 
8

1517670 
3005291

13628983 
C.R.,
12/2000 
9

26246220

189116129

399085891 
P.
J., 31/12/2000 
10 
153363210

302296019
(*) 
1682564911 
P.
J., 5/2000 
(*) This is only the smallest known
sequence...(Gennady Gusev wrote, the 16/6/01: I've
been searching twin AP up to 2.1E9 and found only one AP of length 10  that
Paul Jobling found: N=10 d=153363210
A1=302296019 An=1682564911. So you may take
off the word "known", it is just "the smallest".)
***
One day after, Paul sent the following
& smaller sequence now for K=9:
(81868+i*11362)*11#+1050+1, leading to the
square
372839670 241608570 267854790
189116130 294101010 399085890
320347230 346593450 215362350
Common difference = 11362*11# = 26246220
He can not assure yet that this is the
earliest sequence we are looking for.
***
But the 1/1/2001 Paul closed the
question. How?
"I
have checked my calculations, and I find that the AP of 9 twin primes that I
sent is the smallest. Specifically, I tested (A+Bi).7#+C for A+Bi<2000000,
and found only one AP of length 9: (900553+i*124982).7#
+ 0 +/1. This is the same as (81868+i*11362)*11#+1050+/1."
Then, the absolute earlier solution to the
Emperor question is:
372839669 
241608569 
267854789 

372839670 
241608570 
267854790 

372839671 
241608571 
267854791 
189116129 
294101009 
399085889 
ß1 
189116130 
294101010 
399085890 
+1ŕ 
189116131 
294101011 
399085891 
320347229 
346593449 
215362349 

320347230 
346593450 
215362350 

320347231 
346593451 
215362351 
A problem supposedly
closed 725 years later... not a bad problem...
***
