Problems & Puzzles: Puzzles

Puzzle 168.  Primes such that SOD² = POD

Jason Earls was studying the sequence of primes such that the square of the sum of its digits (SOD) is equal to the product of its digits (POD).

Example: 145451 because (1+4+5+4+5+1)² = 20² = 4x5x4x5

In particular Jason was interested in the palprimes.

I found the earliest: 11172427111.

Questions: 

Can you find the next 9 palprimes of this sequence?

Solution:

Jim Fougeron sent (2/2/02):

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Felice Russo also found (8/2/02) all the primes found by Fougeron except the 711511111115117  because he jumped it on purpose supposing that all primes must start in "1".

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An interesting follow up sent by Patrick De Geest is this one:

"...Since sum and product are 'commutative' operations the prime digit-rearrangements also show up making these a bit less original palprimes. Jim Fougeron's list shows in fact only four distinct palprimes :

1. 11172427111 (CR's entry)
2. 1112184812111
3. 111751111157111
4. 11141114441114111

All the other palprimes are digital permutations of these four. Therefore, Carlos, maybe you extend the puzzle's life by asking a second question requesting for nine 'original' palprimes which satisfy the given property ?..."

19 digits: 1111112424242111111
21 digits: 111122112242211221111
23 digits: 11111141511411514111111
25 digits: 1111111711711171171111111 and 1111122111154511112211111
27 digits: no solution found
29 digits: no solution found
31 digits: 1111111111151154511511111111111
33 digits: no solution found
35 digits: 11111111174111111411111147111111111
37 digits: 1111111121711112114112111171211111111
39 digits: no solution found
41 digits: no solution found
43 digits: 1111111121111111112184812111111111211111111
45 digits: no solution found
47 digits: 11111111111111111111157475111111111111111111111
49 digits: no solution found
51 digits: 111111111111211111212211242112212111112111111111111