Problems & Puzzles: Puzzles

Puzzle 467. Primes in the last two digits JM Bergot sent one more nice puzzle: One notices that from 4211 to 4297 that there is a sequence of 15 primes, producing15 increasing primes for the last two digits. 4211 11 4217 17 4219 19 4229 29 4231 31 4241 41 4243 43 4253 53 4259 59 4261 61 4271 71 4273 73 4283 83 4289 89 4297 97 Q1. Is there a longer run of primes having this property? The two-digits-primes need not be consecutive, just increasing. Q2. Is there a longer run of primes producing strictly consecutive primes for the last two digits? Q3. Is there a longer run of primes producing primes for the last two digits, no matter if increasing or not?

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J. K. Andersen wrote:

Q2. Let p be 78314167738064529047713 or 163027495131423420474913.
Then there is a prime 17-tuplet from p to p+66, discovered by Peter Leikauf & Joerg Waldvogel in 2001:
http://anthony.d.forbes.googlepages.com/ktuplets.htm#largest17

The 17 primes end in 17 consecutive two-digit primes:
13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79.

Q3 I don't know whether the same two-digit prime is allowed to occur more than once in Q3. For the first p, the next primes are p+154 and p+166 which end in the already occurred two-digit primes 67 and 79.

For the second p, the next prime is p+84 which ends in the non-consecutive
increasing prime 97. Then comes p+340 which ends in the already occurred 53.

Below are primes above 100 which start a longer run than any smaller prime when decreasing and repeated two-digit primes are allowed, but leading zeros in 03, 05, 07 are disallowed.

run:prime
1: 113
2: 131
3: 167
6: 311
8: 353
11: 613
12: 3313
15: 4211
16: 2102311
17: 17812037
19: 24492119
24: 100338311
25: 579934153
26: 2664266359
27: 2778258229
28: 27529465441
29: 191845798043
30: 287377190629
32: 3065219827559
35: 3664533202453

The primes in the run of 35 are 3664533200000 + n, for n = 2453, 2511, 2513, 2529, 2543, 2567, 2579, 2619, 2679, 2717, 2741, 2747, 2783, 2823, 2831, 2879, 2919, 2937, 3059, 3119, 3131, 3141, 3143, 3161, 3167, 3311, 3347, 3413, 3417, 3437, 3453, 3459, 3483, 3519, 3537.

Here is a non-minimal case of 8 consecutive primes all ending in the SAME two-digit prime 11:
405345364672084109891525651445494532273037052272813939092607711 + n, for n = 0, 100, 300, 400, 600, 700, 900, 1000. Divisibility by 3 makes 1000 the smallest possible span for 8 such primes

Carlos Rivera wrote:

The prime 910935911 starts a run of 9 consecutive primes such that we get the first two digits primes in order from 11 to 41. What is the prime (if it exists) that starts a run of the first 21 two digits primes in order from 11 to 97?.

J. K Andersen wrote about this my question:

If the digits before the last two must be the same each time then it is not possible. The 16 primes from 11 to 71 cover all values modulo 11, so at least one of the 16 numbers will be divisible by 11. If the digits are allowed to be different then the k-tuple conjecture says there will be infinitely many cases, but it looks hard to find one.

If the first prime ends with 11 and the digits before the last two must be the same then the longest possible sequence is 15 primes ending in 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67. The first 2 cases start at 140245881111654813611 and 227209370616659726411. The following prime does not end in 71 in those cases.

The 20 primes from 13 to 97 form an admissible tuple so the k-tuple conjecture predicts infinitely many cases where the preceding digits are the same. Earlier I mentioned two 17-tuples ending in the primes from 13 to 79. That looks hard to beat and I am not trying.

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Torjjörn Alm wrote:

Q3: Checking all primes up to about 4200000 (the first 300000), the following
longer sequences were found:

19 numbers:
3484729 3484759 3484783 3484829 3484837 3484841 3484861 3484879 3484889 3484903 3484907 3484913 3484931 3484967 3484979 3484997 3485011 3485047 3485059

18 numbers:
171541 171553 171559 171571 171583 171617 171629 171637 171641 171653 171659 171671 171673 171679 171697 171707 171713 171719

17 numbers:
2967317 2967323 2967329 2967331 2967337 2967343 2967347 2967353 2967359 2967361 2967373 2967379 2967383 2967389 2967397 2967403 2967407
This sequence also contains 15 numbers with the 2 last digits increasing.

16 numbers:
161641 161659 161683 161717 161729 161731 161741 161743 161753 161761 161771 161773 161779 161783 161807 161831

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Enoch Haga wrote:

Q3). There seems to be an infinite number of longer runs producing primes for the last two digits. Among the first are a run of 16 -- 2102267-2102531; 17 -- 17812037-17812343; 19 -- 65750819-65751143; 24 -- 100338311-100338713 (includes all primes excepting 37, 43, 53, 67, 73, 89, just 15 different primes in all).

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Farideh Firoozbakht wrote:

Q3: Yes there is. 2208751703 to 2208752323 that there is a sequence
of 32 primes, producing 32 primes (18 distinct primes) for the last two digits.

Primes from 2208751703 to 2208752323:

2208751703, 2208751759, 2208751807, 2208751817, 2208751819,
2208751823, 2208751837, 2208751841, 2208751859, 2208751861,
2208751871, 2208751873, 2208751903, 2208751913, 2208751967,
2208751997, 2208752003, 2208752053, 2208752113, 2208752131,
2208752141, 2208752173, 2208752179, 2208752197, 2208752213,
2208752219, 2208752237, 2208752243, 2208752267, 2208752279,
2208752303 & 2208752323.

32 primes for the last two digits:

3, 59, 7, 17, 19, 23, 37, 41, 59, 61, 71, 73, 3,13, 67, 97, 3,
53, 13, 31, 41, 73, 79, 97, 13, 19, 37, 43, 67, 79, 3 & 23.

18 distinct primes for the last two digits:
3, 7, 13, 17, 19, 23, 31, 37, 41, 43, 53, 59, 61, 67, 71, 73, 79 & 97.

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