Problems & Puzzles: Puzzles

 Puzzle 506. Reversible primes again Enoch Haga sent the following puzzle: Find consecutive reversible primes such that the reversed primes are descending in order. Examples 3 primes: 71 73 79 bent to 97 37 17 4 primes 1231 1237 1249 1259 bent to 9521 9421 7321 1321 Q1. Can you find a bent sequence with 5 or more consecutive primes?

Contributions came from Dmitry Kamenetsky, JC Rosa, Farideh Firoozbakht & J. K. Andersen

***

Dmitry wrote:

2, 3, 5, 7, 11, 13, 17
9597192601, 9597192613, 9597192643, 9597192653, 9597192667, 9597192677, 9597192679
70289935351, 70289935361, 70289935363, 70289935367, 70289935369, 70289935379, 70289935489

I also found many bent sequences of 5 and 6 consecutive primes. The ones with the largest starting prime are:

99984879541, 99984879583, 99984879593, 99984879667, 99984879677
99878011231, 99878011271, 99878011273, 99878011297, 99878011319, 99878011339

***

JC Rosa wrote:

I have only found solutions for 5 and 6 consecutive primes.

5 consecutive primes:
P=1958711, 1958731,1958753,1958773,1958777
R=7778591,3778591,3578591,1378591,1178591

6 consecutive primes:
P=18974551, 18974561,18974581,18974591,18974623,18974629
R=92647981,32647981,19547981,18547981,16547981,15547981

I have tested P up to 7*10^9

***

Farideh wrote:

Smallest sequence of k consecutive emirps with the requested property for k = 5, 6 & 7:

k = 5 ;

1958711,1958731,1958753,1958773,1958777
1178591,1378591,3578591,3778591,7778591

k = 6 ;

18974551,18974561,18974581,18974591,18974623,18974629
15547981,16547981,18547981,19547981,32647981,92647981

k = 7 ;

9597192601,9597192613,9597192643,9597192653,9597192667,9597192677,9597192679
1062917959,3162917959,3462917959,3562917959,7662917959,7762917959,9762917959

Smallest sequence of k consecutive emirps p1, p2, ..., pk such that
reversal(p1) > reversal(p2) > ... > reversal(pk) for k = 2, 3, 4 & 5 :

k = 2 ;

739,743
937,347

k = 3 ;

739,743,751
937,347,157

k = 4 ;

92959,92987,92993,93001
95929,78929,39929,10039

k = 5 ;

30357479,30357487,30357517,30357553,30357571
97475303,78475303,71575303,35575303,17575303

Andersen wrote:

The first case of 5 consecutive primes:
1958711 1958731 1958753 1958773 1958777 bent to
7778591 3778591 3578591 1378591 1178591

First case of 6:
18974551 18974561 18974581 18974591 18974623 18974629 bent to
92647981 32647981 19547981 18547981 16547981 15547981

First case of 7:
9597192601 9597192613 9597192643 9597192653 9597192667 9597192677 9597192679
9762917959 7762917959 7662917959 3562917959 3462917959 3162917959 1062917959

First case of 5 primes for which the concatenation is also a reversible prime:
9684668131 9684668153 9684668173 9684668183 9684668207 bent to
7028664869 3818664869 3718664869 3518664869 1318664869

***

 Records   |  Conjectures  |  Problems  |  Puzzles