Problems & Puzzles: Puzzles

 

 

Problems & Puzzles: Puzzles

Puzzle 1247 The 24 permutations related to the four ending digits

On Oct 28, G. L. Honaker, Jr. sent the following interesting puzzle.

 
Disregarding the primes 2 & 5, all the primes end in one of the following four digits: 1, 3, 7 & 9

With these four ending digits you may form 24 permutations, shown in the box below:

{1,3,7,9} {1,3,9,7} {1,7,3,9} {1,7,9,3} {1,9,3,7} {1,9,7,3}
{3,1,7,9} {3,1,9,7} {3,7,1,9} {3,7,9,1} {3,9,1,7} 
{3,9,7,1} 
{7,1,3,9} {7,1,9,3} {7,3,1,9} {7,3,9,1} {7,9,1,3} {7,9,3,1}
{9,1,3,7}
 {9,1,7,3} {9,3,1,7} {9,3,7,1} {9,7,1,3} {9,7,3,1}

G. L. Honaker, Jr. asked sometime ago the following question: "what is the smallest prime p in which the frequency of the primes less or equal than p for each ending digit in a given permutation, appearing in decreasing order?"

Example: the permutation {3,7,1,9} is such that the frequency of the primes below or equal to 53 appearing in decreasing order:

f(3)=5: (3, 13, 23, 43, 53)
f(7)=4: (7, 17, 37, 47)
f(1)=3: (11, 31, 41)
f(9)=2: (19, 29)

Honaker, Jr. himself found the solution to three of these permutations (shown in bold blue letters in the box).

Mike Keith has found 16 more solutions (shown in blod black letters in the box).

Here are all the already found solutions.

Count of primes per ending digit form 2 to SPP
for each permutation
Permutation Smallest prime 1 3 7 9
Sol. No. (SPP) Sum (1,3,7,9) π(SPP)*
1 3719 53 3 5 4 2 14 16
2 3791 239 11 14 13 12 50 52
3 7319 347 17 18 19 13 67 69
4 7391 619 26 29 30 27 112 114
5 7931 3529 121 122 125 123 491 493
6 7913 3581 124 123 127 125 499 501
7 3179 6781 219 221 218 213 871 873
8 3971 10949 330 335 331 332 1328 1330
9 7139 23431 654 653 656 644 2607 2609
10 9731 72269 1778 1790 1791 1792 7151 7153
11 3917 176081 3993 4017 3992 3996 15998 16000
12 3197 960389 18901 18947 18898 18899 75645 75647
13 9371 6263539 107350 107457 107432 107458 429697 429699
14 1739 45845791 691626 691618 691625 691419 2766288 2766290
15 1379 45853453 691733 691730 691729 691532 2766724 2766726
16 7193 646488749 8404882 8404863 8405723 8404864 33620332 33620334
17 1397 716407555481 6821352349 6821352348 6821338021 6821340492 27285383210 27285383212
18 9317  5140571155981 45524194118 45524206173 45524194117 45524209220 182096803628 182096803630
19 9137 5150660154191 45610372205 45610372204 45610365360 45610385683 182441495452 182441495454

* π(SPP) counted using the web page: https://t5k.org/nthprime/

 

The Nineteen permutations already solved are in bold Black or Blue letters in the box above.
The Five pending solutions are the rest of permutations in bold Red in the same box.

Q1. Can you confirm the correctness of the first 19 solutions?

Q2. Send your solutions for the 5 pending permutations. 


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