Puzzle 1156 1738967

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Puzzle 1156 1738967

Metin Sariyar sent the following nice puzzle.
1738967 is the smallest prime p such that p, R(p), p+2, R(p+2), R(p)+2, R(R(p)+2), p+6, R(p+6), R(p)+6, R(R(p)+6) are all distinct primes, where "R" is reversal.

Q. Find the next example or greater examples... (except the reversal 7698371).

Ten distinct primes below;

p=1738967, R(p)=7698371

p+2=1738969, R(p+2)=9698371

R(p)+2=7698373, R(R(p)+2)=3738967

p+6=1738973, R(p+6)=3798371

R(p)+6=7698377, R(R(p)+6)=7738967

Duriing the week from 31 Dec-6 Jan 2023, contributions came from Michael Branicky, V.F.Izquierdo, Paul Cleary, Emmanuek Vantieghem, JM Rebert, Giorgos Kalogeropoulos, Oscar Volpatti

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Michael wrote:
I found 10 more, the lowest being 10397429591, listed as p and the other 9 distinct primes:

p R(p) p+2 R(p+2) R(p)+2 R(R(p)+2) p+6 R(p+6) R(p)+6 R(R(p)+6)

10397429591, 19592479301, 10397429593, 39592479301, 19592479303, 30397429591, 10397429597, 79592479301, 19592479307, 70397429591
11285254361, 16345258211, 11285254363, 36345258211, 16345258213, 31285254361, 11285254367, 76345258211, 16345258217, 71285254361

12450298871, 17889205421, 12450298873, 37889205421, 17889205423, 32450298871, 12450298877, 77889205421, 17889205427, 72450298871

13754285471, 17458245731, 13754285473, 37458245731, 17458245733, 33754285471, 13754285477, 77458245731, 17458245737, 73754285471

15621281777, 77718212651, 15621281779, 97718212651, 77718212653, 35621281777, 15621281783, 38718212651, 77718212657, 75621281777

16345258211, 11285254361, 16345258213, 31285254361, 11285254363, 36345258211, 16345258217, 71285254361, 11285254367, 76345258211

17458245731, 13754285471, 17458245733, 33754285471, 13754285473, 37458245731, 17458245737, 73754285471, 13754285477, 77458245731

17889205421, 12450298871, 17889205423, 32450298871, 12450298873, 37889205421, 17889205427, 72450298871, 12450298877, 77889205421

18028789607, 70698782081, 18028789609, 90698782081, 70698782083, 38028789607, 18028789613, 31698782081, 70698782087, 78028789607

19592479301, 10397429591, 19592479303, 30397429591, 10397429593, 39592479301, 19592479307, 70397429591, 10397429597, 79592479301

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Vicente wrote:

The 3 next terms:

Prime[472296718]=10397429591
Prime[510718112]=11285254361
Prime[560943917]=12450298871

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Paul wrote:

Here are the next 10 solutions below 10^9 the prime. using the same format as the original.

10397429591
19592479301
10397429593
39592479301
19592479303
30397429591
10397429597
79592479301
19592479307
70397429591

11285254361
16345258211
11285254363
36345258211
16345258213
31285254361
11285254367
76345258211
16345258217
71285254361

12450298871
17889205421
12450298873
37889205421
17889205423
32450298871
12450298877
77889205421
17889205427
72450298871

13754285471
17458245731
13754285473
37458245731
17458245733
33754285471
13754285477
77458245731
17458245737
73754285471

15621281777
77718212651
15621281779
97718212651
77718212653
35621281777
15621281783
38718212651
77718212657
75621281777

16345258211
11285254361
16345258213
31285254361
11285254363
36345258211
16345258217
71285254361
11285254367
76345258211

17458245731
13754285471
17458245733
33754285471
13754285473
37458245731
17458245737
73754285471
13754285477
77458245731

17889205421
12450298871
17889205423
32450298871
12450298873
37889205421
17889205427
72450298871
12450298877
77889205421

18028789607
70698782081
18028789609
90698782081
70698782083
38028789607
18028789613
31698782081
70698782087
78028789607

19592479301
10397429591
19592479303
30397429591
10397429593
39592479301
19592479307
70397429591
10397429597
79592479301

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Emmanuel Vantieghem wrote

The next p is 10397429591, generating the primes
10397429593,19592479301,10397429597,39592479301,79592479301,19592479303,30397429591,19592479307,70397429591.

The next p are :
11285254361
12450298871
13754285471
14858185841 palindrome ! ! !
15621281777
15934643951 palindrome ! ! !
18028789607
19592479301

In my opinion, the palindrome cases may be considered as "not serious" because the generated primes are not all different.

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JM Rebert wrote

Least solution with d-digit primes

d [p, R(p), p+2, R(p+2), R(p)+2, R(R(p)+2), p+6, R(p+6), R(p)+6, R(R(p)+6)]

7 [1738967, 7698371, 1738969, 9698371, 7698373, 3738967, 1738973, 3798371, 7698377, 7738967]

11 [10397429591, 19592479301, 10397429593, 39592479301, 19592479303, 30397429591, 10397429597, 79592479301, 19592479307, 70397429591]

12 [107525284877, 778482525701, 107525284879, 978482525701, 778482525703, 307525284877, 107525284883, 388482525701, 778482525707, 707525284877]

13 [1000935120761, 1670215390001, 1000935120763, 3670215390001, 1670215390003, 3000935120761, 1000935120767, 7670215390001, 1670215390007, 7000935120761]

14 [10002077423207, 70232477020001, 10002077423209, 90232477020001, 70232477020003, 30002077423207, 10002077423213, 31232477020001, 70232477020007, 70002077423207]

15 [100109920557527, 725755029901001, 100109920557529, 925755029901001, 725755029901003, 300109920557527, 100109920557533, 335755029901001, 725755029901007, 700109920557527]

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Giorgos wrote:

Next examples that produce 10 primes are:
10397429591, 11285254361, 12450298871, 13754285471, 14858185841, 15621281777,

15934643951, 16345258211, 17458245731, 17889205421, 18028789607, 19592479301, 70698782081,

72490958537, 73582143077, 73585909427, 75635753657, 77034128537, 77718212651, 79595759597,

107525284877, 111673062377, 112156891367, 150587412737, 154962341537, 706917339257,

710331323747, 718710337817, 718733017817, 735143269451, 737214785051, 747323133017,

752933719607, 763198651211, 773260376111, 778482525701...

Also, the first number that has additional property p+8 to be prime is 77718212651

Finally, the next two numbers have 2 additional properties: p+8 and R(p+8) are primes

1258234713641 and 1566048381341

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Oscar wrote:

Let me restate Metin's example as follows:

p is an emirp, with reversal q = R(p);

next k primes after p are numbers p+a_1, ..., p+a_k, and they are emirps too;

moreover, numbers q+a_1, ..., q+a_k are emirps too (maybe the first k primes after q, maybe not).

So we have 2+4*k distinct primes.

Metin's example is the case k=2, a_1=2, a_2=6.

The first solution is p=1738967.

Next solution is p=10397429591, with emirp pairs:

10397429591, 19592479301,

10397429593, 39592479301,

10397429597, 79592479301,

19592479303, 30397429591,

19592479307, 70397429591.

Metin chose a_1 and a_2 as small as possible.

Without such constraint, the solution with minimal p is a_1=16, a_2=34, p=37813, giving emirp pairs:

37813, 31873,

37831, 13873,

37847, 74873,

31891, 19813,

31907, 70913.

For k=3, the solution with minimal p is a_1=16, a_2=48, a_3=70, p=30742711, giving emirp pairs:

30742711, 11724703,

30742727, 72724703,

30742759, 95724703,

30742781, 18724703,

11724719, 91742711,

11724751, 15742711,

11724773, 37742711.

It would be nicer to choose a_1=2, a_2=6, a_3=8, as small as possible, but I found no such solution.

My best solution so far is a_1=10, a_2=16, a_3=22, p=142061411851, giving emirp pairs:

142061411851, 158114160241,

142061411861, 168114160241,

142061411867, 768114160241,

142061411873, 378114160241,

158114160251, 152061411851,

158114160257, 752061411851,

158114160263, 362061411851.

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