Problems & Puzzles: Puzzles

Puzzle 1009. The tallest pyramid of Emirps

Carlos Rivera produced the following 38-levels pyramid of emirps, constructed this way:

1. Start with any of the eight emirp of two digits, for the first level

2. Insert one and any digit of the set {0-9} in any internal position that produces an emirp for the next level.

3. Repeat 2 until no more levels can be produced.

 Level Emirp 1 31 2 311 3 3011 4 30161 5 390161 6 3901561 7 39016561 8 390163561 9 3901763561 10 39061763561 11 390619763561 12 3906189763561 13 39046189763561 14 390461897563561 15 3904601897563561 16 39046018975463561 17 390460189075463561 18 3904601890755463561 19 37904601890755463561 20 357904601890755463561 21 3547904601890755463561 22 35547904601890755463561 23 355479004601890755463561 24 3655479004601890755463561 25 36556479004601890755463561 26 365564790046018907554636561 27 3655647900460189075546365671 28 36556479004601890755463656701 29 365564769004601890755463656701 30 3655647659004601890755463656701 31 36556476590046018907554643656701 32 365564765900460189075354643656701 33 3655647659004601890753504643656701 34 36556476590046018907535204643656701 35 365564765900460618907535204643656701 36 3655647659004606189073535204643656701 37 36556476590046061890473535204643656701 38 365586476590046061890473535204643656701

* in red the digit inserted in each level

But not sure this is the largest pyramid of emirps ever possible.

Q. Can you send a larger one pyramid of emirps?

During the week 118-24, July, 2020, contributions came from Simon Cavegn, Oscar Volpatti, Paul Cleary, Ray Opao, Emmanuel Vantieghem.

***

Simon wrote:

If I coded correctly:
There are no longer solutions.
There are 4 solution of same length.
Here the 3 other solutions:

31
3*1*1
3*0*11
301*6*1
3*9*0161
3901*5*61
3901*6*561
39016*3*561
3901*7*63561
390*6*1763561
39061*9*763561
39061*8*9763561
390*4*6189763561
390461897*5*63561
39046*0*1897563561
39046018975*4*63561
390460189*0*75463561
39046018907*5*5463561
3*7*904601890755463561
3*5*7904601890755463561
35*4*7904601890755463561
3*5*547904601890755463561
355479*0*04601890755463561
3*6*55479004601890755463561
3655*6*479004601890755463561
36556479004601890755463*6*561
36556479004601890755463656*7*1
365564790046018907554636567*0*1
3655647*6*9004601890755463656701
36556476*5*9004601890755463656701
365564765900460189075546*4*3656701
365564765900460189075*3*54643656701
36556476590046018907535*0*4643656701
36556476590046018907535*2*04643656701
365564765900460*6*18907535204643656701
365*4*564765900460618907535204643656701
36545647659004606*9*18907535204643656701
36545647659004*3*606918907535204643656701

31
3*1*1
3*0*11
301*6*1
3*9*0161
3901*5*61
3901*6*561
39016*3*561
3901*7*63561
390*6*1763561
39061*9*763561
39061*8*9763561
390*4*6189763561
390461897*5*63561
39046*0*1897563561
39046018975*4*63561
390460189*0*75463561
39046018907*5*5463561
3*7*904601890755463561
3*5*7904601890755463561
35*4*7904601890755463561
3*5*547904601890755463561
355479*0*04601890755463561
3*6*55479004601890755463561
3655*6*479004601890755463561
36556479004601890755463*6*561
36556479004601890755463656*7*1
365564790046018907554636567*0*1
3655647*6*9004601890755463656701
36556476*5*9004601890755463656701
365564765900460189075546*4*3656701
365564765900460189075*3*54643656701
36556476590046018907535*0*4643656701
36556476590046018907535*2*04643656701
365564765900460*6*18907535204643656701
365564765900460618907*3*535204643656701
36556476590046061890*4*73535204643656701
36556476590046061890473535*2*204643656701

31
3*1*1
3*0*11
301*6*1
3*9*0161
3901*5*61
3901*6*561
39016*3*561
3901*7*63561
390*6*1763561
39061*9*763561
39061*8*9763561
390*4*6189763561
390461897*5*63561
39046*0*1897563561
39046018975*4*63561
390460189*0*75463561
39046018907*5*5463561
3*7*904601890755463561
3*5*7904601890755463561
35*4*7904601890755463561
3*5*547904601890755463561
355479*0*04601890755463561
3*6*55479004601890755463561
3655*6*479004601890755463561
36556479004601890755463*6*561
36556479004601890755463656*7*1
365564790046018907554636567*0*1
3655647*6*9004601890755463656701
36556476*5*9004601890755463656701
365564765900460189075546*4*3656701
365564765900460189075*3*54643656701
36556476590046018907535*0*4643656701
36556476590046018907535*2*04643656701
3655647659004601890753520464365670*1*1
3655647659004*2*60189075352046436567011
365564765900426*4*0189075352046436567011
36556476590042640*8*189075352046436567011

***

Oscar wrote:

We can't beat your pyramid of emirps, but we can get a tie.

Starting with 37, 73, 79, and 97, we can't exceed 30 levels.
Starting with 17, 71, and 11 (palprime), we can't exceed 33 levels.
Starting with 13 and 31, we can't exeed 38 levels.
There are eight distinct 38-levels pyramids; curiously, they differ on at most six levels.

In the following, the new digit inserted in each level is usually marked in red;
if it is inserted near an identical digit, they are both marked in green.

LEVELS 3-4: two paths

Path 1 (Rivera)
311

3011
30161
390161

Path 2
311
3911
39161
390161

LEVELS 35-38: four paths

Path 1 (Rivera)
36556476590046018907535204643656701

365564765900460618907535204643656701
3655647659004606189073535204643656701
36556476590046061890473535204643656701
365586476590046061890473535204643656701

Path 2 (differs from Rivera's on level 38 only)

36556476590046018907535204643656701

365564765900460618907535204643656701
3655647659004606189073535204643656701
36556476590046061890473535204643656701
365564765900460618904735352204643656701

Path 3 (differs from From Rivera's on levels 36 to 38 only)

36556476590046018907535204643656701
365564765900460618907535204643656701
3654564765900460618907535204643656701
36545647659004606918907535204643656701
365456476590043606918907535204643656701

Path 4:
36556476590046018907535204643656701
365564765900460189075352046436567011
3655647659004260189075352046436567011
36556476590042640189075352046436567011
365564765900426408189075352046436567011

So, this pyramid is the farthest from the given example:

31

311

3911

39161

390161

3901561

39016561

390163561

3901763561

39061763561

390619763561

3906189763561

39046189763561

390461897563561

3904601897563561

39046018975463561

390460189075463561

3904601890755463561

37904601890755463561

357904601890755463561

3547904601890755463561

35547904601890755463561

355479004601890755463561

3655479004601890755463561

36556479004601890755463561

365564790046018907554636561

3655647900460189075546365671

36556479004601890755463656701

365564769004601890755463656701

3655647659004601890755463656701

36556476590046018907554643656701

365564765900460189075354643656701

3655647659004601890753504643656701

36556476590046018907535204643656701

365564765900460189075352046436567011

3655647659004260189075352046436567011

36556476590042640189075352046436567011

365564765900426408189075352046436567011

***

Paul wrote:

I haven’t done a full audit trail back to the 2 digit emirp, but I can confirm the largest number is the one you have published.  Here is a full list of all the 39 digit emirps that can be reduced back to 2 digits removing 1 digit at a time, sorted by magnitude.

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365586476590046061890473535204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900460618904735352204643656701

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365564765900426408189075352046436567011

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

365456476590043606918907535204643656701

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

110765634640253570981804624009567465563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535709819606340095674654563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402535374098160640095674685563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563

107656346402253537409816064009567465563.

***

Ray wrote:

I brute-forced all 2-digit emirps using some Python code and I came up with 4 distinct values for the 38th level (including the one you have):

365456476590043606918907535204643656701
365586476590046061890473535204643656701
365564765900460618904735352204643656701
365564765900426408189075352046436567011

But I did not find anything beyond the 38th level. I assumed that starting with 2-digit emirps limited the possible starting points. So, I tried starting with 3-digit emirps and ended up with 37 levels that ended with the SAME 4 distinct values. I tried starting with 4-digit emirps and resulted with 36 levels that still ended with the SAME 4 distinct values.

It was only when I started with 5-digit emirps that I ended with a different set of  2 distinct values at the 36th level. Starting with 6-digit emirps resulted in another set of 2 distinct values, still at the 36th level.

I didn't try beyond 6-digit emirps.

***

Emmanuel wrote:

a) Starting with a two-digit emirp gives no taller solution.

b) The first taller solution (trapezoid) starts with an eight-digit emirp :

97508899
977508899
9775008899
97750080899
978750080899
9787500804899
97875008084899
978175008084899
9728175008084899
97281750080814899
972817509080814899
9728175090808154899
97281753090808154899
972817953090808154899
9723817953090808154899
97238179530900808154899
972381799530900808154899
9723817995309100808154899
97238179953091009808154899
972381799530910098081548999
9672381799530910098081548999
96723817995309100980810548999
967238179995309100980810548999
9672381799953091009808010548999
96723817999530910098080105489929
936723817999530910098080105489929
9396723817999530910098080105489929
93967238179995309100938080105489929
939672381799953091009380801054839929
9396723817999530910093808010564839929
93906723817999530910093808010564839929
939067423817999530910093808010564839929
9390674238179995308910093808010564839929
93906742381799953089100938080105648396929
939067423817999530891009381080105648396929
9390674238617999530891009381080105648396929
93906742386179995308910093810980105648396929
939067842386179995308910093810980105648396929
9390607842386179995308910093810980105648396929
(39 emirps).

***

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