Problems & Puzzles: Puzzles

Puzzle 700 p & q primes such that ... Emmanuel Vantieghem sent the following nice puzzle Find pairs or non-palindromic primes (p & q) such that all of the following 12 integers are 12 distinct primes: p, q, R(p), R(q), p&q, q&p, p&R(q), R(q)&p, R(p)&q, q&R(p), R(p)&R(q), R(q)&R(p) One example is this one: p=155599, q=752197. p 155599 q 752197 R(p) 995551 R(q) 791257 p&q 155599752197 q&p 752197155599 p&R(q) 155599791257 R(q)&p 791257155599 q&R(p) 752197995551 R(p)&q 995551752197 R(p)&R(q) 995551791257 R(q)&R(p) 791257995551 Emmanuel sent three more examples. Q. Find the first 10 examples (ordered by p&q).

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Contributions came from Giovanni Resta, Jud McCranie, Emmanuel Vantieghem, W. Edwin Clark, Jahngeer Kholdi & Farideh Firoozbakht and Vicente Felipe Izquierdo

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Giovanni wrote:

Clearly if (p,q) is a solution, also (p,rev(q)),  (rev(p),q), (rev(p),rev(q)) and the 4 other pairs obtained swapping p and q are solutions, since all lead to the same set of 12 numbers.

So I confined the search to pairs for which p<q, p<rev(p) and q<rev(q).

For p,q < 10^7 there are 40 such pairs.
The 10 smallest ones, in terms of  the largest component are:

186007 302647
162391 700459
379909 705247
155599 752197
324143 1089359
12119 1324619
365479 1513591
32299 1548181
72689 1568519
1150927 1589689

Later he added

I found the smallest pair of consecutive primes
which produce a set of 12 primes. They are:
(1724280032963, 1724280033023)

If one of the two consecutive primes is allowed to be
palindromic (thus reducing the resulting set to 7 primes), then the smallest solution is (70280108207, 70280108219)

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Jud wrote:

Attached are 50 solutions to puzzle 700.  When listing a reversible prime, I list the smaller number.  The lines are p, q pairs, with q>p.  The list is sorted in order of increasing q.

186007 302647
162391 700459
379909 705247
155599 752197
324143 1089359
12119 1324619
365479 1513591
32299 1548181
72689 1568519
1150927 1589689
74897 1632209
11057 1699307
32939 1845017
13553 1852013
1025209 1862317
1587503 1866593
1311853 1873357
1716499 1879069
197647 1913467
1509887 3023747
16699 3248347
1996229 3289283
301459 3295657
3081523 3319609
1390507 3403339
14923 3513427
1207387 3547909
162017 3637367
336079 3806119
313517 3830429
15383 3841787
3276073 3844747
1353019 3923287
3823649 7020509
31259 7141919
30319 7241029
3117053 7382399
1294369 7576669
7006187 9163079
1860569 9280199
340909 10009603
1004981 10037309
79379 10064081
1963369 10079053
1415207 10209491
7665439 10321639
1576703 10396091
1821649 10548883
3305833 10714831
1266731 10854071

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Emmanuel wrote:

I computed the first 53 sets of twelve primes.  Each such a set consits of four emirps, say p, q, r(=R(p)), s(=R(q))   and the eight concatenations

p&q, p&s, q&p, q&r, s&p, r&q, s&r, r&s.  All concatenations have at most twelve digits.

Of the concatenations I retain the smallest one and I write it down with a space instead of the symbol '&'.  

  149 37805387
  1031 34240589 
  1069 11025337 
  1091 33734849 
  11057 1699307 
  113 170699843  
  115949633 179 
  12119 1324619 
  1212722131 13 
  1223 38711249 
  1300969741 13 
  1302807491 17 
  13 1371020383 
  13 1613060191 
  13258151 3023 
  13 3057444307 
  1348937531 17 
  13553 1852013 
  136562731 337 
  13 7420441099 
  1397512849 79 
  1471 19007017 
  1475759227 37 
  14923 3513427 
  15383 3841787 
  1548181 32299 
  155063791 709 
  155599 752197 
  1568519 72689 
  162391 700459 
  1632209 74897 
  16699 3248347 
  16863563 3527 
  17 7433830367 
  1845017 32939 
  186007 302647 
  1906713169 37 
  191480039 389 
  19852747 9349 
  30319 7241029 
  308544617 347 
  3110744863 79 
  31259 7141919 
  3140189179 37 
  3198372427 37 
  3469023367 79 
  347 386363249 
  347 703185257 
  3526253167 37 
  359 925232549 
  3719 38487629 
  37 9144955489 
  379909 705247
 

This list should be 'complete' in the sense that, if someone finds a concatenation (with twelve digits or less) not in this list, then either the concatenation was not the smallest in the corresponding set of twelve primes, or the set does not contain twelve different primes (a mistake that I made first).

Of course, Ii is possible that I made mistakes.  Therefore it would be interesting to see this results doublechecked.

Besides, I conjecture that for every emirp  p  there exist an emirp  q  that will allow the construction (as above) of 12 different emirps.  There is a heuristic argument for that. Anyhow, I verified the conjecture for  p < 1000.

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Clark wrote:

First, one notices that if (p,q) is a solution then so are all eight of the
primes pairs: { (p,q),  (p,R(q)), (R(p),q), (R(p),R(q), (q,p), (R(q),p),  (q,R(p)), (R(q),R(p))}.
[One can interpret this as an orbit of the dihedral group of order 8 acting on
the set of pairs.]

In all I found 22 distinct orbits of such pairs. Here are representatives of the 10 orbits
with smallest p&q values that I  found: I  would not be surprised if I missed some.

(1 ) 149,37805387
(2 ) 1069,11025337
(3 ) 11057,1699307
(4 ) 113,170699843
(5 ) 115949633,179
(6 ) 12119,1324619
(7 ) 1223,38711249
(8 ) 13258151,3023
(9 ) 13553,1852013
(10) 136562731,337
 

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Jahngeer Kholdi & Farideh Firoozbakht wrote:

It is clear if pair (p, q)  is a solution, then all 11 other pairs   (p, R(q)),

(q, p), ... , (R(p), R(q)) have the same property. 

 

The solutions are as follows:

 

a1.   p = 1031 & q = 34240589

a2.   p = 1031 & q = 98504243

b1.   p = 1069 & q = 11025337

b2.   p = 1069 & q = 73352011

c1.   p = 1091 & q = 33734849

c2.   p = 1091 & q = 94843733

b3.   p = 11025337 & q = 1069

d1.   p = 113 & q = 170699843

d2.   P = 113 & q = 348996071

e1.   p = 12119 & q = 1324619

f1.    p = 1212722131 & q = 13

g1.   p = 1223 & q = 38711249

g2.   p = 1223 & q = 94211783

h1.   p = 1300969741 & q = 13

a3.   p = 1301 & q = 34240589
a4.   p = 1301 & q = 98504243
***
Vicente wrote:

The least primes what I found.
 
149, 37805387
149, 78350873
941, 37805387
941, 78360873
1031, 34240589
1069, 11025337
1069, 73352011
1091, 33734849
1223, 38711249
1301, 34240589
1471, 19007017
1471, 71070091
1741, 19007017
1741, 77107091
1901, 33734849
3023, 13258151
3023, 15185231

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