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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.
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|
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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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