Problems & Puzzles: Puzzles

Puzzle 852. Primes and convolution Vic Bold sent the following nice puzzle related to an operation defined by him as "convolution" The convolution C(N) of an integer N is an operation made over the decimal digits of N, according to the following rule: "Just reverse the order of the digits and multiply them in pairs" Example N=135 then C(135)=135x531=5+9+5=19 Now on, we will be only interested in primes P such that P+C(P) generates another prime Q: Examples P=11, P+C(P)=11+2=13 P=17, P+C(P)=17+14=31 You may verify that the following four primes 11, 13, 17 & 19 are consecutive and generate primes adding to each its respective convolution: {11,13,17,19} ->{13, 19, 31, 37} Q1. Find a larger set of consecutive primes P that generate primes by convolution. You may verify that the following primes form a chain of 5 primes where the last four are generated by convolution after the first one: 11 -> 13 -> 19 -> 37 -> 79 Q2. Find larger chains of primes generated by convolution. You may verify that both 19 & 31 generate the same prime 37, by convolution: {19, 31}->37 Q3 Find larger sets of primes that generate by convolution the same prime. BTW I made a fast search and found the following examples for Q1, Q2 & Q3: Q1. A set of 10 consecutive primes that generate primes by convolution: {909865877, 909865889, 909865919, 909865937, 909865949, 909865961, 909866011, 909866021, 909866029, 909866063} Q2. A chain of 10 primes generated by convolution after the first one: 2859289943 -> 2859290303 -> 2859290381 -> 2859290579 -> 2859290813 -> 2859290957 -> 2859291191 -> 2859291403 -> 2859291509 -> 2859291649 Q3. 7 primes P that generate the same prime Q:  {682605689, 682605697, 682605757, 682605809, 682605817, 682605841, 682605901} -> 682606009

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Contributions came from jan van Delden and Emmanuel Vantieghem

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Jan wrote:

Q1:

 

11 consecutive primes (same as given, but starts 1 prime earlier..)

909865867, 909865877, 909865889, 909865919, 909865937, 909865949,

909865961, 909866011, 909866021, 909866029, 909866063

 

12 consecutive primes:

70555626979, 70555626989, 70555626997, 70555627001, 70555627031, 70555627033,

70555627069, 70555627097, 70555627141, 70555627171, 70555627183, 70555627201

 

Q2:

 

Longer chains have 12 digits or more.

 

Q3:

 

8 primes that generate 68207071553:

68207071259  68207071267 68207071291 68207071319

68207071327  68207071351 68207071403 68207071411

 

9 primes that generate 6829098862097: (*)

6829098861659, 6829098861667, 6829098861691, 6829098861719,  6829098861727,  6829098861743, 6829098861751, 6829098861803, 6829098861811

 

(*) This solution might not be the smallest. I also searched for these type of solutions where the “prime”-condition is dropped. The first digits of the numbers involved seem to settle on a specific pattern, which I used to speed up the process of finding this 13-digit solution. The same pattern emerges if one imposes the “prime”-condition (but it is more clear).

 

If one uses 13-digit numbers (not necessarily prime):

26 numbers generate 6829981000443:

6829980999699, 6829980999759, 6829980999767, 6829980999775, 6829980999783, 6829980999791, 6829980999819, 6829980999827, 6829980999835, 6829980999843, 6829980999851, 6829980999903, 6829980999911, 6829981000198, 6829981000258, 6829981000266, 6829981000274, 6829981000282, 6829981000290, 6829981000318, 6829981000326, 6829981000334, 6829981000342, 6829981000350, 6829981000402, 6829981000410

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Emmanuel wrote:

Q3.  Here are eight primes that map to the prime  682790658527  :

        682790658089, 682790658149, 682790658157, 682790658181, 682790658217, 682790658233, 682790658241, 682790658301.

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