Problems & Puzzles: Puzzles

Puzzle 913. The last two digits of a prime is a prime. Vic Bold sent the following nice puzzle: Find a large non trivial set of successive primes such that for each, the last two (by the right) digits is a prime number (03 & 07 counts as primes) Vic sent the following example, 15 successive primes: 14713  14717  14723  14731  14737  14741  14747  14753  14759  14767  14771  14779  14783  14797  14813 By my side I (CR) found 32 successive of these primes starting at 2208751703 with only 18 distinct prime terminations. Q1. Send your largest set of successive primes like that. Q2. Send the smallest set of successive primes like that, such that also contains at least once each of the first possible two digits primes (23): 03, 07, 11, ... 97. Q3. Redo Q1 for the last three digits.

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Contribution came from Jan van Delden and Carlos Rivera.

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Jan wrote on Feb 19, 2018:

Q1:

I decided to start my routine looking for primes having at least 2 digits.
So I decided not to count a solution starting with prime 7 (a chain of length 22).

The longest chain of consecutive primes I found consisted of 41 primes.

 

The following chain has a total of 22 distinct tail-primes (of 2 digits) ij green:

 

8455391581789,8455391581807,8455391581823,8455391581831

8455391581837,8455391581883,8455391581913,8455391581943

8455391581961,8455391581979,8455391581997,8455391582003

8455391582017,8455391582047,8455391582059,8455391582117

8455391582119,8455391582161,8455391582167,8455391582173

8455391582203,8455391582219,8455391582237,8455391582279

8455391582317,8455391582371,8455391582389,8455391582479

8455391582567,8455391582597,8455391582641,8455391582707

8455391582717,8455391582731,8455391582753,8455391582767

8455391582783,8455391582843,8455391582867,8455391582873

8455391582929

 

The first chain of 41 consecutive primes, with 20 distinct tail-primes starts at:

 

4241713956013

 

Q2:

 

I used a brute force attack and didn’t find a chain with 23 distinct tail-primes, I searched until 3000*2^32.

The first chain having 22 distinct tail-primes has length 26 and starts at:

 

113314970383

 

The most efficient chain having 22 distinct tail-primes has length 23 and starts at:

 

1584656304031

 

[I couldn’t find a chain having 22 distinct tail-primes with length 22]

 

Q3:

 

I decided to start my routine looking for prime having at least 3 digits.
The longest chain starts at p=101 and has length 143.
If one decides to start counting from prime 7 the length would be 165.
Both results are trivial and hard to beat.

 

The longest chain I found consist of 53 consecutive primes, starting at:

 

976911936037

 

All 3-digit tail-primes are distinct.

 

I searched quite a bit further (until 2000*2^32) but couldn’t find a longer chain

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Rivera wrote of Feb 20, 2018:

246246017 is the start of a run of 51 consecutive prime numbers such the last three digits of each is also a prime number. This is the largest run before 2^32. See this curio.

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Giovanni Resta wrote on March 9, 2018:

I extended the search for Puzzle 931.

For Q1, I have found the next two record-length sequences
longer than 41. They have length 44 and 45 and start
at 16256855416117 and 105120615024773, respectively.

For Q2, I found several sequences that contains all the 23
possible primes.
The earliest two have length 28 and 26, and start at
16985591466307 and 17065972501531, respectively.

I also found the first prime that starts a run of 23
primes all with distinct endings. This run is embedded into
a longer sequence. The sequence of these 23 primes is:

90442568793803, 90442568793817, 90442568793937, 90442568793947,
90442568793961, 90442568794013, 90442568794031, 90442568794073,
90442568794153, 90442568794159, 90442568794211, 90442568794219,
90442568794229, 90442568794241, 90442568794267, 90442568794271,
90442568794307, 90442568794343, 90442568794379, 90442568794397,
90442568794423, 90442568794483, 90442568794489.

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