Problems & Puzzles: Puzzles

Puzzle 511. Primes reversing N, N^2, & N^3, ..., N^K Frank Rubin, old friend of mine and these pages, poses the following nice puzzle: Find an integer N such that when you reverse the digits of N you get a prime, when you reverse the digits of N^2 you get a prime, and when you reverse the digits of N^3 you get a prime. Solvers are encouraged to try N^4 and beyond. C. Rivera, in a preliminary fast examination of the difficulty of this puzzle, found the following results: K N | Rev(N^I)=Prime for I=1 to K 2 14 3 3112 4 11182 5 1109311 6 110218462 So, the new challenge is to get N values for K>6.

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Contributions came from J. K. Andersen & Farideh Firoozbakht

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Andersen wrote:

The smallest solutions for K = 2 to 7 are:
K N
2 14
3 325
4 3244
5 3244
6 110218462
7 32149366346

They are also listed in http://www.research.att.com/~njas/sequences/A165696

I found no case of K = 8 below 10^13, but 72 with K = 7. 11 were below 10^12:
32149366346, 32610710725, 99685879657, 101740496633, 314898426203,
318616427746, 319609794146, 321020993636, 322871279992, 323428923685,
324097084138.

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Farideh wrote:

The smallest number N such that all seven numbers Rev(N^I), I <= 7 are prime is 32149366346 and also smallest prime number with this property is 101740496633.

Please see http://primes.utm.edu/curios/page.php?number_id=9558  .

For k = 6 instead of 11028462 must be 110218462.
Also for k = 3, 4 & 5 your table doesn't give us the smallest N.

KN | Rev(N^I)=Prime for I=1 to K
2,14
3,3112
4,11182
5,1109311
6,110218462

According to the terms of the sequences A165696 and A165698 for the smallest number N and smallest prime P such that all the k numbers Rev(N^I) and all the k numbers Rev(P^I) I <= k are prime we have two following tables.

KSmallest N | Rev(N^I)=Prime for I=1 to K
1,2
2,14
3,325
4,3244
5,3244
6,110218462
7,32149366346

KSmallest prime P | Rev(P^I)=Prime for I=1 to K
1,2
2,37
3,3121
4,10429
5,10282339
6,9835884797
7,101740496633

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