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):

The first 13 primes satisfying puzzle 168 are:

11721412711
17112421171
1112184812111
1128114118211
111751111157111
117511111115711
151711111117151
711511111115117
11141114441114111
11141141414114111
11144111411144111
11411411411411411

***

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".

***

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 ?..."

***

Felice Russo has found (22/2/02) the 5th of the list asked by Patrick.

5. 1111112424242111111 (Russo)

***

J.C. Rosa sent (25/2/02) the following results to the Patrick's follow up:

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

I am not absolutely sure, in each case, to have found the smallest.

***

 Records   |  Conjectures  |  Problems  |  Puzzles