Problems & Puzzles: Puzzles

 Puzzle 1102 Consecutive primes ending in the same digit JM Bergot sent the following nice puzzle:   For three consecutive primes all ending in the same last digit and summing to a prime: 19+29+59=107 29+59+79=167 59+79+89=227 79+89+109=277 89+109+139=337 109+139+179=397 =139+149+179=467 Q. Using only consecutive primes ending in the same digit, can you  find more than seven consecutive successes producing a prime, as above?

During the week 11-17 set. 2022, contributions came from Giorgos Kalogeropoulos, Emmanuel Vantieghem, Paolo Lava, Michael Branicky, Adam Stinchcombe, Jean-Marc Rebert, Gennady Gusev, Oscar Volpatti

***

Giorgios wrote:

The first primes that produce 8 consecutive successes are:
{80768473, 173905759, 388683929, 615723877, 651775987...}

The least prime that produce 10  consecutive successes is 3388711097

3388711097 + 3388711127 + 3388711147 = 10166133371
3388711127 + 3388711147 + 3388711357 = 10166133631
3388711147 + 3388711357 + 3388711367 = 10166133871
3388711357 + 3388711367 + 3388711477 = 10166134201
3388711367 + 3388711477 + 3388711597 = 10166134441
3388711477 + 3388711597 + 3388711607 = 10166134681
3388711597 + 3388711607 + 3388711727 = 10166134931
3388711607 + 3388711727 + 3388711747 = 10166135081
3388711727 + 3388711747 + 3388711957 = 10166135431
3388711747 + 3388711957 + 3388712057 = 10166135761

***

Emmanuel wrote:

May I point out a printing error in the announcement of puzzle 1102 ?
I think :
...
109+139+179= 397
...
should become
...
109+139+149= 397
...

On the other hand, this is what I found :
__________________________________

Solutions with eight sums :

80768473 + 80768503 + 80768693 = 242305669
80768503 + 80768693 + 80768713 = 242305909
80768693 + 80768713 + 80768783 = 242306189
80768713 + 80768783 + 80768903 = 242306399
80768783 + 80768903 + 80768923 = 242306609
80768903 + 80768923 + 80769163 = 242306989
80768923 + 80769163 + 80769173 = 242307259
80769163 + 80769173 + 80769203 = 242307539

173905759 + 173906009 + 173906189 = 521717957
173906009 + 173906189 + 173906269 = 521718467
173906189 + 173906269 + 173906279 = 521718737
173906269 + 173906279 + 173906329 = 521718877
173906279 + 173906329 + 173906419 = 521719027
173906329 + 173906419 + 173906459 = 521719207
173906419 + 173906459 + 173906479 = 521719357
173906459 + 173906479 + 173906569 = 521719507

615723877 + 615723917 + 615723967 = 1847171761
615723917 + 615723967 + 615724097 = 1847171981
615723967 + 615724097 + 615724147 = 1847172211
615724097 + 615724147 + 615724357 = 1847172601
615724147 + 615724357 + 615724367 = 1847172871
615724357 + 615724367 + 615724427 = 1847173151
615724367 + 615724427 + 615724477 = 1847173271
615724427 + 615724477 + 615724597 = 1847173501

***

Paolo wrote:

The first set with 8 consecutive primes with the same LSD is 80768473, 80768503, 80768693, 80768713, 80768783, 80768903, 80768923, 80769163, 80769173, 80769203:

The second set with 8 consecutive primes with the same LSD is 173905759, 173906009, 173906189, 173906269, 173906279, 173906329, 173906419, 173906459, 173906479, 173906569

***

Michael wrote:

I found 8 primes in a row arising from sums of the following triples of consecutive same-last-digit primes:
(80768473, 80768503, 80768693), (80768503, 80768693, 80768713), (80768693, 80768713, 80768783), (80768713, 80768783, 80768903), (80768783, 80768903, 80768923), (80768903, 80768923, 80769163), (80768923, 80769163, 80769173), (80769163, 80769173, 80769203)
(173905759, 173906009, 173906189), (173906009, 173906189, 173906269), (173906189, 173906269, 173906279), (173906269, 173906279, 173906329), (173906279, 173906329, 173906419), (173906329, 173906419, 173906459), (173906419, 173906459, 173906479), (173906459, 173906479, 173906569)
(615723877, 615723917, 615723967), (615723917, 615723967, 615724097), (615723967, 615724097, 615724147), (615724097, 615724147, 615724357), (615724147, 615724357, 615724367), (615724357, 615724367, 615724427), (615724367, 615724427, 615724477), (615724427, 615724477, 615724597)
(1921001711, 1921001891, 1921001941), (1921001891, 1921001941, 1921002071), (1921001941, 1921002071, 1921002191), (1921002071, 1921002191, 1921002241), (1921002191, 1921002241, 1921002271), (1921002241, 1921002271, 1921002401), (1921002271, 1921002401, 1921002491), (1921002401, 1921002491, 1921002541)

Exactly 9 in a row:
(5833686161, 5833686211, 5833686241), (5833686211, 5833686241, 5833686281), (5833686241, 5833686281, 5833686421), (5833686281, 5833686421, 5833686541), (5833686421, 5833686541, 5833686611), (5833686541, 5833686611, 5833686631), (5833686611, 5833686631, 5833686701), (5833686631, 5833686701, 5833686931), (5833686701, 5833686931, 5833686941)
(13417145509, 13417145579, 13417145719), (13417145579, 13417145719, 13417145839), (13417145719, 13417145839, 13417145879), (13417145839, 13417145879, 13417145899), (13417145879, 13417145899, 13417145969), (13417145899, 13417145969, 13417146019), (13417145969, 13417146019, 13417146299), (13417146019, 13417146299, 13417146389), (13417146299, 13417146389, 13417146499)
(15122395817, 15122395927, 15122396117), (15122395927, 15122396117, 15122396327), (15122396117, 15122396327, 15122396587), (15122396327, 15122396587, 15122396867), (15122396587, 15122396867, 15122396977), (15122396867, 15122396977, 15122396987), (15122396977, 15122396987, 15122397047), (15122396987, 15122397047, 15122397127), (15122397047, 15122397127, 15122397307)
(45241956413, 45241956433, 45241956523), (45241956433, 45241956523, 45241956563), (45241956523, 45241956563, 45241956623), (45241956563, 45241956623, 45241956703), (45241956623, 45241956703, 45241956713), (45241956703, 45241956713, 45241956733), (45241956713, 45241956733, 45241956863), (45241956733, 45241956863, 45241956923), (45241956863, 45241956923, 45241956943)

Exactly 10 in a row (N.B.: for last-digit 7, happens before exactly 9 in a row):
(3388711097, 3388711127, 3388711147), (3388711127, 3388711147, 3388711357), (3388711147, 3388711357, 3388711367), (3388711357, 3388711367, 3388711477), (3388711367, 3388711477, 3388711597), (3388711477, 3388711597, 3388711607), (3388711597, 3388711607, 3388711727), (3388711607, 3388711727, 3388711747), (3388711727, 3388711747, 3388711957), (3388711747, 3388711957, 3388712057)

***

So far the best I have found (many length 7, quite a few length 8, no 9's) a length 10 sequence of prime sums using as summands the sequence of 12 primes

3388711097,3388711127,3388711147,3388711357,3388711367,
3388711477,3388711597,3388711607,3388711727,3388711747,
3388711957,3388712057

...

Follow up: 11 primes for 9 prime summands starting at 5833686161, 12 primes for 10 summands starting at 4106800597 and again at 3388711097.

***

Jean-Marc wrote:

Q. Using only consecutive primes ending in the same digit, can you  find more than seven consecutive successes producing a prime, as above?

c : record of consecutive successes.
p0 : starting prime of a run of primes ending with the same digit. 1) p0 <= 10^11

c p0
1 11
2 41
3 101
6 601
7 41131
8 1921001711 *
9 5833686161 *

1 13
2 103
3 4513
5 12973
6 1482263
7 2146433
8 80768473 *
9 45241956413 *

6 7
7 4861807
8 615723877 *
10 3388711097 *

7 19
8 173905759 *
9 13417145509 *

Example :

1) p0 <= 10^12
a) Primes ending in the same digit 1.

c p0
1 11
1 [11, 31, 41] -> 83 prime
2 [31, 41, 61] -> 133 not prime
c = 1

c p0
2 41
1 [41, 61, 71] -> 173 prime
2 [61, 71, 101] -> 233 prime
3 [71, 101, 131] -> 303 not prime
c = 2

c p0
3 101
1 [101, 131, 151] -> 383 prime
2 [131, 151, 181] -> 463 prime
3 [151, 181, 191] -> 523 prime
4 [181, 191, 211] -> 583 not prime
c = 3

c p0
6 601
1 [601, 631, 641] -> 1873 prime
2 [631, 641, 661] -> 1933 prime
3 [641, 661, 691] -> 1993 prime
4 [661, 691, 701] -> 2053 prime
5 [691, 701, 751] -> 2143 prime
6 [701, 751, 761] -> 2213 prime
7 [751, 761, 811] -> 2323 not prime
c = 6

c p0
7 41131
1 [41131, 41141, 41161] -> 123433 prime
2 [41141, 41161, 41201] -> 123503 prime
3 [41161, 41201, 41221] -> 123583 prime
4 [41201, 41221, 41231] -> 123653 prime
5 [41221, 41231, 41281] -> 123733 prime
6 [41231, 41281, 41341] -> 123853 prime
7 [41281, 41341, 41351] -> 123973 prime
8 [41341, 41351, 41381] -> 124073 not prime
c = 7

8 1921001711
1 [1921001711, 1921001891, 1921001941] -> 5763005543 prime
2 [1921001891, 1921001941, 1921002071] -> 5763005903 prime
3 [1921001941, 1921002071, 1921002191] -> 5763006203 prime
4 [1921002071, 1921002191, 1921002241] -> 5763006503 prime
5 [1921002191, 1921002241, 1921002271] -> 5763006703 prime
6 [1921002241, 1921002271, 1921002401] -> 5763006913 prime
7 [1921002271, 1921002401, 1921002491] -> 5763007163 prime
8 [1921002401, 1921002491, 1921002541] -> 5763007433 prime
9 [1921002491, 1921002541, 1921002641] -> 5763007673 not prime
c = 8

9 5833686161
1 [5833686161, 5833686211, 5833686241] -> 17501058613 prime
2 [5833686211, 5833686241, 5833686281] -> 17501058733 prime
3 [5833686241, 5833686281, 5833686421] -> 17501058943 prime
4 [5833686281, 5833686421, 5833686541] -> 17501059243 prime
5 [5833686421, 5833686541, 5833686611] -> 17501059573 prime
6 [5833686541, 5833686611, 5833686631] -> 17501059783 prime
7 [5833686611, 5833686631, 5833686701] -> 17501059943 prime
8 [5833686631, 5833686701, 5833686931] -> 17501060263 prime
9 [5833686701, 5833686931, 5833686941] -> 17501060573 prime
10 [5833686931, 5833686941, 5833687001] -> 17501060873 not prime
c = 9

b) Primes ending in the same digit 3.

c p0
1 13
1 [13, 23, 43] -> 79 prime
2 [23, 43, 53] -> 119 not prime
%38 = 1

c p0
2 103
1 [103, 113, 163] -> 379 prime
2 [113, 163, 173] -> 449 prime
3 [163, 173, 193] -> 529 not prime
c = 2

c p0
3 4513
1 [4513, 4523, 4583] -> 13619 prime
2 [4523, 4583, 4603] -> 13709 prime
3 [4583, 4603, 4643] -> 13829 prime
4 [4603, 4643, 4663] -> 13909 not prime
c = 3

c p0
5 12973
1 [12973, 12983, 13003] -> 38959 prime
2 [12983, 13003, 13033] -> 39019 prime
3 [13003, 13033, 13043] -> 39079 prime
4 [13033, 13043, 13063] -> 39139 prime
5 [13043, 13063, 13093] -> 39199 prime
6 [13063, 13093, 13103] -> 39259 not prime
c = 5

c p0
6 1482263
1 [1482263, 1482293, 1482343] -> 4446899 prime
2 [1482293, 1482343, 1482413] -> 4447049 prime
3 [1482343, 1482413, 1482443] -> 4447199 prime
4 [1482413, 1482443, 1482583] -> 4447439 prime
5 [1482443, 1482583, 1482743] -> 4447769 prime
6 [1482583, 1482743, 1482763] -> 4448089 prime
7 [1482743, 1482763, 1482773] -> 4448279 not prime
c = 6

c p0
7 2146433
1 [2146433, 2146483, 2146523] -> 6439439 prime
2 [2146483, 2146523, 2146633] -> 6439639 prime
3 [2146523, 2146633, 2146663] -> 6439819 prime
4 [2146633, 2146663, 2146673] -> 6439969 prime
5 [2146663, 2146673, 2146693] -> 6440029 prime
6 [2146673, 2146693, 2146723] -> 6440089 prime
7 [2146693, 2146723, 2146733] -> 6440149 prime
8 [2146723, 2146733, 2146763] -> 6440219 not prime
c = 7

8 80768473
1 [80768473, 80768503, 80768693] -> 242305669 prime
2 [80768503, 80768693, 80768713] -> 242305909 prime
3 [80768693, 80768713, 80768783] -> 242306189 prime
4 [80768713, 80768783, 80768903] -> 242306399 prime
5 [80768783, 80768903, 80768923] -> 242306609 prime
6 [80768903, 80768923, 80769163] -> 242306989 prime
7 [80768923, 80769163, 80769173] -> 242307259 prime
8 [80769163, 80769173, 80769203] -> 242307539 prime
9 [80769173, 80769203, 80769223] -> 242307599 not prime
c = 8

9 45241956413
1 [45241956413, 45241956433, 45241956523] -> 135725869369 prime
2 [45241956433, 45241956523, 45241956563] -> 135725869519 prime
3 [45241956523, 45241956563, 45241956623] -> 135725869709 prime
4 [45241956563, 45241956623, 45241956703] -> 135725869889 prime
5 [45241956623, 45241956703, 45241956713] -> 135725870039 prime
6 [45241956703, 45241956713, 45241956733] -> 135725870149 prime
7 [45241956713, 45241956733, 45241956863] -> 135725870309 prime
8 [45241956733, 45241956863, 45241956923] -> 135725870519 prime
9 [45241956863, 45241956923, 45241956943] -> 135725870729 prime
10 [45241956923, 45241956943, 45241956973] -> 135725870839 not prime
%16 = 9

c) Primes ending in the same digit 7.

c p0
6 7
1 [7, 17, 37] -> 61 prime
2 [17, 37, 47] -> 101 prime
3 [37, 47, 67] -> 151 prime
4 [47, 67, 97] -> 211 prime
5 [67, 97, 107] -> 271 prime
6 [97, 107, 127] -> 331 prime
7 [107, 127, 137] -> 371 not prime
%35 = 6

c p0
7 4861807
1 [4861807, 4861877, 4861907] -> 14585591 prime
2 [4861877, 4861907, 4861957] -> 14585741 prime
3 [4861907, 4861957, 4862027] -> 14585891 prime
4 [4861957, 4862027, 4862047] -> 14586031 prime
5 [4862027, 4862047, 4862087] -> 14586161 prime
6 [4862047, 4862087, 4862107] -> 14586241 prime
7 [4862087, 4862107, 4862437] -> 14586631 prime
8 [4862107, 4862437, 4862717] -> 14587261 not prime
c = 7

8 615723877
1 [615723877, 615723917, 615723967] -> 1847171761 prime
2 [615723917, 615723967, 615724097] -> 1847171981 prime
3 [615723967, 615724097, 615724147] -> 1847172211 prime
4 [615724097, 615724147, 615724357] -> 1847172601 prime
5 [615724147, 615724357, 615724367] -> 1847172871 prime
6 [615724357, 615724367, 615724427] -> 1847173151 prime
7 [615724367, 615724427, 615724477] -> 1847173271 prime
8 [615724427, 615724477, 615724597] -> 1847173501 prime
9 [615724477, 615724597, 615724687] -> 1847173761 not prime
c = 8

10 3388711097
1 [3388711097, 3388711127, 3388711147] -> 10166133371 prime
2 [3388711127, 3388711147, 3388711357] -> 10166133631 prime
3 [3388711147, 3388711357, 3388711367] -> 10166133871 prime
4 [3388711357, 3388711367, 3388711477] -> 10166134201 prime
5 [3388711367, 3388711477, 3388711597] -> 10166134441 prime
6 [3388711477, 3388711597, 3388711607] -> 10166134681 prime
7 [3388711597, 3388711607, 3388711727] -> 10166134931 prime
8 [3388711607, 3388711727, 3388711747] -> 10166135081 prime
9 [3388711727, 3388711747, 3388711957] -> 10166135431 prime
10 [3388711747, 3388711957, 3388712057] -> 10166135761 prime
11 [3388711957, 3388712057, 3388712177] -> 10166136191 not prime
c = 10

d) Primes ending in the same digit 9.

c p0
7 19
1 [19, 29, 59] -> 107 prime
2 [29, 59, 79] -> 167 prime
3 [59, 79, 89] -> 227 prime
4 [79, 89, 109] -> 277 prime
5 [89, 109, 139] -> 337 prime
6 [109, 139, 149] -> 397 prime
7 [139, 149, 179] -> 467 prime
8 [149, 179, 199] -> 527 not prime
c = 7

8 173905759
1 [173905759, 173906009, 173906189] -> 521717957 prime
2 [173906009, 173906189, 173906269] -> 521718467 prime
3 [173906189, 173906269, 173906279] -> 521718737 prime
4 [173906269, 173906279, 173906329] -> 521718877 prime
5 [173906279, 173906329, 173906419] -> 521719027 prime
6 [173906329, 173906419, 173906459] -> 521719207 prime
7 [173906419, 173906459, 173906479] -> 521719357 prime
8 [173906459, 173906479, 173906569] -> 521719507 prime
9 [173906479, 173906569, 173906599] -> 521719647 not prime
c = 8

9 13417145509
2 [13417145579, 13417145719, 13417145839] -> 40251437137 prime
3 [13417145719, 13417145839, 13417145879] -> 40251437437 prime
4 [13417145839, 13417145879, 13417145899] -> 40251437617 prime
5 [13417145879, 13417145899, 13417145969] -> 40251437747 prime
6 [13417145899, 13417145969, 13417146019] -> 40251437887 prime
7 [13417145969, 13417146019, 13417146299] -> 40251438287 prime
8 [13417146019, 13417146299, 13417146389] -> 40251438707 prime
9 [13417146299, 13417146389, 13417146499] -> 40251439187 prime
10 [13417146389, 13417146499, 13417146539] -> 40251439427 not prime
c = 9

***

I have found solutions > 7 for every digit 1, 3, 7, 9.
```n=10
3388711097+3388711127+3388711147=10166133371
3388711127+3388711147+3388711357=10166133631
3388711147+3388711357+3388711367=10166133871
3388711357+3388711367+3388711477=10166134201
3388711367+3388711477+3388711597=10166134441
3388711477+3388711597+3388711607=10166134681
3388711597+3388711607+3388711727=10166134931
3388711607+3388711727+3388711747=10166135081
3388711727+3388711747+3388711957=10166135431
3388711747+3388711957+3388712057=10166135761

n=8
173905759+173906009+173906189=521717957
173906009+173906189+173906269=521718467
173906189+173906269+173906279=521718737
173906269+173906279+173906329=521718877
173906279+173906329+173906419=521719027
173906329+173906419+173906459=521719207
173906419+173906459+173906479=521719357
173906459+173906479+173906569=521719507

n=8
80768473+80768503+80768693=242305669
80768503+80768693+80768713=242305909
80768693+80768713+80768783=242306189
80768713+80768783+80768903=242306399
80768783+80768903+80768923=242306609
80768903+80768923+80769163=242306989
80768923+80769163+80769173=242307259
80769163+80769173+80769203=242307539

n=9
5833686161+5833686211+5833686241=17501058613
5833686211+5833686241+5833686281=17501058733
5833686241+5833686281+5833686421=17501058943
5833686281+5833686421+5833686541=17501059243
5833686421+5833686541+5833686611=17501059573
5833686541+5833686611+5833686631=17501059783
5833686611+5833686631+5833686701=17501059943
5833686631+5833686701+5833686931=17501060263
5833686701+5833686931+5833686941=17501060573```

***

Oscar wrote:

I've checked all primes p < 2.7*10^12, finding up to 12 consecutive successes.

n = 8 successes:
80768473   242305669
80768503   242305909
80768693   242306189
80768713   242306399
80768783   242306609
80768903   242306989
80768923   242307259
80769163   242307539
80769173
80769203

n = 10 successes:
3388711097   10166133371
3388711127   10166133631
3388711147   10166133871
3388711357   10166134201
3388711367   10166134441
3388711477   10166134681
3388711597   10166134931
3388711607   10166135081
3388711727   10166135431
3388711747   10166135761
3388711957
3388712057

n = 11 successes:
189231092669   567693278357
189231092839   567693278707
189231092849   567693278917
189231093019   567693279187
189231093049   567693279437
189231093119   567693279737
189231093269   567693280057
189231093349   567693280267
189231093439   567693280457
189231093479   567693280577
189231093539   567693280697
189231093559
189231093599

n = 12 successes:
1370683686511   4112051060003
1370683686731   4112051060303
1370683686761   4112051060443
1370683686811   4112051060683
1370683686871   4112051061023
1370683687001   4112051061353
1370683687151   4112051061563
1370683687201   4112051061683
1370683687211   4112051061863
1370683687271   4112051062243
1370683687381   4112051062573
1370683687591   4112051063063
1370683687601
1370683687871

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