Problems & Puzzles: Puzzles

Puzzle 895. K consecutive quartets of consecutive primes with different ending digit Vic Bold sent the following nice puzzle: Find K consecutive quartets of consecutive primes with different ending digit. He sent the following examples: For K=1: 11, 13, 17, 19 For K=2: 47, 53, 59, 61 53, 59, 61, 67 For K=5 191, 193, 197, 199 193, 197, 199, 211 197, 199, 211, 223 199, 211, 223, 227 211, 223, 227, 229  It was for me not very hard to find a larger example; K=14 2732087209 2732087227 2732087231 2732087263 2732087227 2732087231 2732087263 2732087279 2732087231 2732087263 2732087279 2732087317 2732087263 2732087279 2732087317 2732087321 2732087279 2732087317 2732087321 2732087333 2732087317 2732087321 2732087333 2732087359 2732087321 2732087333 2732087359 2732087447 2732087333 2732087359 2732087447 2732087461 2732087359 2732087447 2732087461 2732087483 2732087447 2732087461 2732087483 2732087489 2732087461 2732087483 2732087489 2732087537 2732087483 2732087489 2732087537 2732087551 2732087489 2732087537 2732087551 2732087563 2732087537 2732087551 2732087563 2732087569 Q. Send your largest example.

---

Jan van Delden wrote on Feb 13, 2018:

I found K=18 defined by the following sequence of 21 consecutive primes:

 

5783272917893, 5783272917901, 5783272917997, 5783272918009,

5783272918043, 5783272918051, 5783272918067, 5783272918109,

5783272918133, 5783272918141, 5783272918187, 5783272918219,

5783272918223, 5783272918231, 5783272918277, 5783272918289,

5783272918343, 5783272918441, 5783272918447, 5783272918459,

5783272918483

 

I searched from 11 until 2000*2^32 and found the following distribution:

 

K   Number of solutions

0   115632207239

1   9335856719

2   2511383931

3   679145191

4   184869118

5   51090778

6   13879708

7   3800046

8   1043105

9   289000

10 78929

11 21590

12 5945

13 1647

14 428

15 134

16 40

17 11

18 1

 

On moving from K to K+1 and comparing frequencies, I would have guessed we would have encountered 2 or 3 sequences of size 18.

 

I also counted the fractions belonging to the order of appearance of the ending digits of two consecutive primes (in the given range above).

As displayed in the directed Graph:

 

So the probability of arriving at digit 3, starting at 1 is estimated by 0.0715. It does indicate that patterns like [1,3,7,9,..] should occur more often.
If one adds these probabilities, either for the arrows pointing away from a vertex or pointing towards a vertex the result is 0.25 as is to be expected.

One could try to compute an estimate of the probability that a certain pattern emerges, I did, but I’m not convinced this is legitimate.

***

---