Problems & Puzzles: Puzzles

 Puzzle 1110 Triplets of consecutive primes Paolo Lava sent the following nice puzzle: Let p, q, r be three consecutive primes. I am looking for triplets such that all the following numbers are primes: a = p + q + r b = R(p) + R(q) + R(r) R(a) R(b) R(a + b) R(R(a) + R(b)) R(a + b + R(a) + R(b)) where R(k) is the reverse of the integer k. I have found two triplets of this kind: (3112301, 3112303, 3112313) and (3216911, 3216931, 3216937). (3112301, 3112303, 3112313) a = 3112301 + 3112303 + 3112313 = 9336917 is prime b = 1032113 + 3032113 + 3132113 = 7196339 is prime R(a) = 7196339 is prime (by the way is equal to b) R(b) = 9336917 is prime (by the way is equal to a)  R(a + b) = R(9336917 + 7196339) = R(16533256) = 65233561 is prime R(R(a) + R(b)) = R(7196339 + 9336917) = R(16533256) = 65233561 is prime (by the way is equal to R(a+ b)) R(a + b + R(a) + R(b)) = R(9336917 + 7196339 + 7196339 + 9336917) = R(33066512) = 21566033 is prime. (3216911, 3216931, 3216937) a = 3216911 + 3216931 + 3216937 =  9650779  is prime b = 1196123 + 1396123 + 7396123 = 9988369  is prime R(a) = 9770569  is prime R(b) = 9638899  is prime  R(a + b) = R(9650779 + 9988369) = R(19639148) = 84193691  is prime R(R(a) + R(b)) = R(9770569 + 9638899) = R(19409468) = 86490491  is prime. R(a + b + R(a) + R(b)) = R(9650779 + 9988369 + 9770569 + 9638899) = R(39048616) = 61684093 is prime. Q. Other triplets like these ones?

During the week 6-12 November, 2022, contributions came from Michael Branicky, Paul Cleary, Giorgos Kalogeropoulos, Oscar Volpatti, Emmanuel Vantieghem

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Michael wrote:

The next such triples are:
(30022012301, 30022012303, 30022012313),
(30402740591, 30402740593, 30402740623),
(31203120001, 31203120017, 31203120041),
(31429920703, 31429920713, 31429920721),
(31780372801, 31780372817, 31780372831),
(31857330773, 31857330853, 31857330881),
(32089237231, 32089237241, 32089237307),
(32151813613, 32151813653, 32151813713),
(32460200671, 32460200683, 32460200693)

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Paul wrote:

Here are a few more triplets.

{30022012301,30022012303,30022012313}

a = 30022012301 + 30022012303 + 30022012313 = 90066036917 is a prime

b = 10321022003 + 30321022003 + 31321022003 = 71963066009 is a prime

R(a) = 71963066009 is a prime (by the way is equal to b)

R(b) = 90066036917 is a prime (by the way is equal to a)

R(a + b) = R(71963066009 + 90066036917) = R(162029102926) = 629201920261 is prime

R(R(a) + R(b) = R(71963066009 + 90066036917) = R(162029102926) = 629201920261 is prime (by the way is equal to R(a + b))

R(a + b +R(a) + R(b)) = R(90066036917 + 71963066009 + 71963066009 + 90066036917) = R(324058205852) = 258502850423 is prime

{30402740591,30402740593,30402740623}

a = 30402740591 + 30402740593 + 30402740623 = 91208221807 is a prime

b = 19504720403 + 39504720403 + 32604720403 = 91614161209 is a prime

R(a) = 91614161209 is a prime (by the way is equal to b)

R(b) = 91208221807 is a prime (by the way is equal to a)

R(a + b) = R(91614161209 + 91208221807) = R(182822383016) = 610383228281 is prime

R(R(a) + R(b) = R(91614161209 + 91208221807) = R(182822383016) = 610383228281 is prime (by the way is equal to R(a + b))

R(a + b +R(a) + R(b)) = R(91208221807 + 91614161209 + 91614161209 + 91208221807) = R(343850804854) = 458408058343 is prime

{31203120001,31203120017,31203120041}

a = 31203120001 + 31203120017 + 31203120041 = 93609360059 is a prime

b = 10002130213 + 71002130213 + 14002130213 = 95006390639 is a prime

R(a) = 95006390639 is a prime (by the way is equal to b)

R(b) = 93609360059 is a prime (by the way is equal to a)

R(a + b) = R(95006390639 + 93609360059) = R(188615750698) = 896057516881 is prime

R(R(a) + R(b) = R(95006390639 + 93609360059) = R(188615750698) = 896057516881 is prime (by the way is equal to R(a + b))

R(a + b +R(a) + R(b)) = R(93609360059 + 95006390639 + 95006390639 + 93609360059) = R(377231501396) = 693105132773 is prime

{31429920703,31429920713,31429920721}

a = 31429920703 + 31429920713 + 31429920721 = 94289762137 is a prime

b = 30702992413 + 31702992413 + 12702992413 = 75108977239 is a prime

R(a) = 75108977239 is a prime (by the way is equal to b)

R(b) = 94289762137 is a prime (by the way is equal to a)

R(a + b) = R(75108977239 + 94289762137) = R(169398739376) = 673937893961 is prime

R(R(a) + R(b) = R(75108977239 + 94289762137) = R(169398739376) = 673937893961 is prime (by the way is equal to R(a + b))

R(a + b +R(a) + R(b)) = R(94289762137 + 75108977239 + 75108977239 + 94289762137) = R(335803517782) = 287715308533 is prime

{31780372801,31780372817,31780372831}

a = 31780372801 + 31780372817 + 31780372831 = 95341118449 is a prime

b = 10827308713 + 71827308713 + 13827308713 = 96481926139 is a prime

R(a) = 96481926139 is a prime (by the way is equal to b)

R(b) = 95341118449 is a prime (by the way is equal to a)

R(a + b) = R(96481926139 + 95341118449) = R(191823044588) = 885440328191 is prime

R(R(a) + R(b) = R(96481926139 + 95341118449) = R(191823044588) = 885440328191 is prime (by the way is equal to R(a + b))

R(a + b +R(a) + R(b)) = R(95341118449 + 96481926139 + 96481926139 + 95341118449) = R(379467077416) = 614770764973 is prime

{31857330773,31857330853,31857330881}

a = 31857330773 + 31857330853 + 31857330881 = 95571992507 is a prime

b = 37703375813 + 35803375813 + 18803375813 = 92310127439 is a prime

R(a) = 92310127439 is a prime (by the way is equal to b)

R(b) = 95571992507 is a prime (by the way is equal to a)

R(a + b) = R(92310127439 + 95571992507) = R(187882119946) = 649911288781 is prime

R(R(a) + R(b) = R(92310127439 + 95571992507) = R(187882119946) = 649911288781 is prime (by the way is equal to R(a + b))

R(a + b +R(a) + R(b)) = R(95571992507 + 92310127439 + 92310127439 + 95571992507) = R(351884138834) = 438831488153 is prime

{32089237231,32089237241,32089237307}

a = 32089237231 + 32089237241 + 32089237307 = 96267711779 is a prime

b = 13273298023 + 14273298023 + 70373298023 = 97919894069 is a prime

R(a) = 97919894069 is a prime (by the way is equal to b)

R(b) = 96267711779 is a prime (by the way is equal to a)

R(a + b) = R(97919894069 + 96267711779) = R(194187605848) = 848506781491 is prime

R(R(a) + R(b) = R(97919894069 + 96267711779) = R(194187605848) = 848506781491 is prime (by the way is equal to R(a + b))

R(a + b +R(a) + R(b)) = R(96267711779 + 97919894069 + 97919894069 + 96267711779) = R(387949274096) = 690472949783 is prime

{32151813613,32151813653,32151813713}

a = 32151813613 + 32151813653 + 32151813713 = 96455440979 is a prime

b = 31631815123 + 35631815123 + 31731815123 = 98995445369 is a prime

R(a) = 98995445369 is a prime (by the way is equal to b)

R(b) = 96455440979 is a prime (by the way is equal to a)

R(a + b) = R(98995445369 + 96455440979) = R(195450886348) = 843688054591 is prime

R(R(a) + R(b) = R(98995445369 + 96455440979) = R(195450886348) = 843688054591 is prime (by the way is equal to R(a + b))

R(a + b +R(a) + R(b)) = R(96455440979 + 98995445369 + 98995445369 + 96455440979) = R(389709801806) = 608108907983 is prime

{32460200671,32460200683,32460200693}

a = 32460200671 + 32460200683 + 32460200693 = 97380602047 is a prime

b = 17600206423 + 38600206423 + 39600206423 = 95800619269 is a prime

R(a) = 95800619269 is a prime (by the way is equal to b)

R(b) = 97380602047 is a prime (by the way is equal to a)

R(a + b) = R(95800619269 + 97380602047) = R(193181221316) = 613122181391 is prime

R(R(a) + R(b) = R(95800619269 + 97380602047) = R(193181221316) = 613122181391 is prime (by the way is equal to R(a + b))

R(a + b +R(a) + R(b)) = R(97380602047 + 95800619269 + 95800619269 + 97380602047) = R(363493430554) = 455034394363 is prime

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Giorgos wrote:

Here are some other triplets of higher orders in the form {p, q, r}

{31408200740514571, 31408200740514593, 31408200740514683},
{3100001301330213001, 3100001301330213043, 3100001301330213053},
{333030001320133321021, 333030001320133321121, 333030001320133321147},
{30132221202132323213321, 30132221202132323213323, 30132221202132323213353},
{3331023223311102021303011, 3331023223311102021303031, 3331023223311102021303047},
{313223233132013231122101131, 313223233132013231122101157, 313223233132013231122101301},
{3132112211210010111200211133121, 3132112211210010111200211133157, 3132112211210010111200211133201},
{301021222003011221133200320311031, 301021222003011221133200320311047, 301021222003011221133200320311101},
{31202022233011211130002201211131230130021, 31202022233011211130002201211131230130107, 31202022233011211130002201211131230130131}

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Oscar wrote:

Let p,q,r be consecutive primes; usually, their base-10 representations have the same length and the same leading digit.
Exceptions are rare and can be checked quickly; up to 2^64, the best exceptional triplet is (2999,3001,3011), with only four primes among the required numbers:
a = 9011,
R(a) = 1109,
R(b) = 89021,
R(R(a)+R(b)) = 03109.

Let's assume that p,q,r have length k>1 and leading digit d.
Both a and b must be emirps, so they both must start and end with digit 1,3,7, or 9.
As b == 3*d mod 10, digit d must be 1,3,7, or 9 too.
But if 7 <= d <= 9, then 2.1*10^k < a < 3*10^k, hence number a starts with digit 2.
So digit d must be 1 or 3 only.

Paolo Lava gave two triplets with k=7 and d=3.
Up to 10^13, there are 187 more solutions.

9 triplets with k=11 and d=3:
(30022012301, 30022012303, 30022012313)
(30402740591, 30402740593, 30402740623)
(31203120001, 31203120017, 31203120041)
(31429920703, 31429920713, 31429920721)
(31780372801, 31780372817, 31780372831)
(31857330773, 31857330853, 31857330881)
(32089237231, 32089237241, 32089237307)
(32151813613, 32151813653, 32151813713)
(32460200671, 32460200683, 32460200693)

6 triplets with k=13 and d=1:
(1065454310981, 1065454311001, 1065454311011)
(1120545750901, 1120545750931, 1120545750941)
(1250121795301, 1250121795311, 1250121795341)
(1258510068001, 1258510068031, 1258510068041)
(1317690323311, 1317690323351, 1317690323401)
(1332123321961, 1332123322001, 1332123322031)

172 triplets with k=13 and d=3:
(3001833158981, 3001833159001, 3001833159007)
...
(3333020030201, 3333020030213, 3333020030233)

Five nice solutions only involve digits 0,1,2, and 3:
(3112301, 3112303, 3112313)
(30022012301, 30022012303, 30022012313)
(3021121013203, 3021121013231, 3021121013233)
(3030323003021, 3030323003023, 3030323003033)
(3333020030201, 3333020030213, 3333020030233)

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Emmanuel wrote:

The next triplets are :
{30022012301,30022012303,30022012313}
{30402740591,30402740593,30402740623}
{31203120001,31203120017,31203120041}
{31429920703,31429920713,31429920721}
{31780372801,31780372817,31780372831}
{31857330773,31857330853,31857330881}
{32089237231,32089237241,32089237307}
{32151813613,32151813653,32151813713}
{32460200671,32460200683,32460200693}
If there is a next triplet, the first element of it will be > 118522211317.

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