Problems & Puzzles: Puzzles

Puzzle 909. Palindrome Friedman prime numbers

Friedman number (FN) is a positive integer which can be written in some non-trivial way using its own digits, together with the symbols + – × / ^ ( ) and concatenation. For example, 25 = 52 and 126 = 21 × 6.

Friedman numbers have been studied by Erich Friedman and others at least since the year 2000. The main results may be seen in the Friedman's page and in A036057.

All Friedman numbers with 6 or fewer digits are known. Here are the FN with 4 or less digits.

``` 25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024,
1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349,
2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592 ,2737, 2916, 3125,
3159, 3281, 3375, 3378, 3685, 3784, 3864, 3972, 4088, 4096, 4106, 4167, 4536, 4624,
4628, 5120, 5776, 5832, 6144, 6145, 6455, 6880, 7928, 8092, 8192, 9025, 9216, 9261.```

For our purposes are enough the following results:

Number of
Digits
Number of
Friedman Numbers
Number of
Friedman prime Numbers *
1 0 0
2 1 0
3 13 2
4 58 1
5 772 46
6 8968 451

* The last column was computed by Carlos Rivera given as valid all the lists of Friedman numbers published in A036057. See my complete list of the Friedman's primes with 6 or less digits here.

A list for the first 9812 Friedman numbers with 6 or less digits may be seen here.

None of the compiled 48 = 2 + 46 Friedman prime numbers with odd quantity of digits is palindrome. Is there any palindrome Friedman prime number? Are there enough Friedman prime numbers?

Fortunately there are infinite quantity of Friedman prime numbers, as was probed by Ron Kaminsky: "The numbers k×1014+19683 = k×106+8+39+0+0+ . . . are Friedman numbers for all k. The numbers of this form are an arithmetic sequence a*n+b where a and b are relatively prime, and therefore, by a well–known theorem of Dirichlet, the sequence contains an infinite number of primes.". Thus, perhaps some of them are palindrome...

Q1. Find the first 5 palindrome Friedman prime numbers?

Q2. If you find Q1 too hard or bored perhaps it is easier to find just 5 palindrome prime numbers of any size, the larger the better.

Contributions came from Jim Howell and Emmanuel Vantieghem

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Jim wrote on 26-1-18:

Using the Ron's formula given in the problem, I have found the following primes for Q2.

3869100000000001000000000019683      k = 38691000000000010 (ten 0's)

3869100000000027200000000019683      k = 38691000000000272

3869100000000068600000000019683      k = 38691000000000686

3869100000000080800000000019683      k = 38691000000000808

386910000000000565000000000019683    k = 386910000000000565

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Emmanuel wrote on 26-1-18:

Q2. The easiest way to construct palprime Friedman numbers is to use Kaminsky's theorem. Here are smallest solutions I could find that way:

3869100000000001000000000019683 = 38691000000000010*10^(6+8)+3^9+0+0+0+0+0+0+0+0
38691000000000000100000000000019683 = 386910000000000001000*10^(6+8)+3^9+0+0+0+0+0+0+0+0
38691000000000000700000000000019683 = 386910000000000007000*10^(6+8)+3^9+0+0+0+0+0+0+0+0
Of course, there might be much smaller solutions, but I couldn't find any.

I used Kaminsky's theorem too in order to find two 'big' ones :

38691*10^1706 + 5*10^855 + 19683 (1711 digits; certified prime by PRIMO in 2h35'22")
38691*10^2048 + 2*10^1026 + 19683 (2053 digits; certified prime by PRIMO in  7h36'18")

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