Problems & Puzzles: Puzzles

Puzzle 943. Primes that remain prime when... Carlos Rivera, poses the following puzzle: Find a prime p such that you can insert in between all the digits of p the digit D for D= 0 to 9, one at a time and the resulting integers remain prime. My best result is p=5769213841 but D goes only from 0 to 8. 5769213841 (this is the seed) The resulting primes are: 5070609020103080401 5171619121113181411 5272629222123282421 5373639323133383431  5474649424143484441 5575659525153585451 5676669626163686461 5777679727173787471 5878689828183888481 Q. Find the minimal prime valid for D= 0 to 9 and, if possible three more alike.

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Contributions came from Carlos Rivera, Shyam Sunder Gupta and Jan van Delden

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Carlos Rivera wrote, Feb 21, 2019:

n.b.:  5769213841 was already found and reported by Emanuel Vantieghem and J. K. Andersen. See my puzzle 557.

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Shyam wrote, Feb 22, 2019:

On testing all primes up to 22801763489, I could not any prime p such that you can insert in between 
all the digits of p the digit D for D= 0 to 9, one at a time and the resulting integers remain prime. 
The result given by you
For p= 5769213841 is the only one for D = 0 to 8.

However I could find one prime p such that you can insert in between all the digits of p the digit D 
for D= 1 to 9, one at a time and the resulting integers remain prime.  The smallest prime p= 5038465463 that remains prime for D=1 to 9 in 5D0D3D8D4D6D5D4D6D3.

Primes produced are 
5101318141615141613,5202328242625242623,5303338343635343633,5404348444645444643, 
5505358545655545653,5606368646665646663,
5707378747675747673,5808388848685848683 and 5909398949695949693

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Jan wrote, Feb 22, 2019:

The following nonprime generates 10 prime solutions: 7691546233. The given solution by Carlos is the smallest solution using digit 0..8.

Given p, the numbers q[d]=p+d*(10^(2k)-1)/99*10, with d in {0,1,..,9} form an arithmetic progression.
For these to be prime it is necessary that (10^(2k)-1)/99*10 is 0 mod r, for r in {2,3,5,7}. From this k=0 mod 3 can be deduced.
For k=12 we get a few extra factors for free, the smallest few are 13,37 and for p there is only 1 free residue mod 11: 1.
One could try and extend this by also considering the 7 free residues mod 17: { 1,2,3,4,5,6,7}.

 

I did spend some time on finding primes p that generate 10 prime solutions of the required form, but didn’t succeed.
I searched k=12, i.e. p has 13 digits.

n.b. 7691546233 was already reported by Emmanuel Vantieghem in our Puzzle 557.

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Giovanni Resta wrote on Saturday 23, Feb 2019:

I forgot to tell you that the solution of Puzzle 943 was already contained in my 2011 answer to Puzzle 72. 

There I wrote that the first composite is 7691546233 and the first prime that generate 10 primes is 1003229774283941. (I also suggested to add it also to Puzzle 557, but probably you missed it.)

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