Problems & Puzzles: Puzzles

Puzzle 1114 Consecutive integers such that... Paolo Lava sent the following nice puzzle: Consider the prime factors of an integer, without multiplicity, put them in ascending order and concatenate. Then check if the resulting number is a prime.   Here is an example of a set of 11 consecutive integers that satisfy the above condition: 13771 -> 47*293 and 47293 is prime; 13772 -> 2^2*11*313 and 211313 is prime; 13773 -> 3*4591 and 34591 is prime; 13774 -> 2*71*97 and 27197 is prime; 13775 -> 5^2*19*29 and 51929 is prime; 13776 -> 2^4*3*7*41 and 23741 is prime; 13777 -> 23*599 and 23599 is prime; 13778 -> 2*83^2 and 283 is prime; 13779 -> 3^2*1531 and 31531 is prime; 13780 -> 2^2*5*13*53 and 251353 is prime; 13781 -> 13781 that is prime itself. 13782 -> 2*3*2297 and 232297 is composite. End of the set. Question: is there a set of consecutive integers greater than 11 that generates primes according to this process.

---

During the week 10-17 December 2022, contributions came from Oscar Volpatti, Giorgos Kalogeropoulos, J-M Rebert

***

Oscar wrote:

I found the first set {a...b} of (exactly) k consecutive integers which generates k primes, for 1 <= k <= 12.

k  a  b
1  27  27
2  36  37
3  11  13
4  16  19
5  21  25
6  162  167
7  12210  12216
8  2  9
9  22572707  22572715
10  312828118  312828127
11  13771  13781
12  173246558990  173246559001

Case k=12, numbers generated:

173246558989 -> 74186235221 = 7*151*70185653
173246558990 -> 2524171886539, prime
173246558991 -> 31125033209719, prime
173246558992 -> 213431490011, prime
173246558993 -> 1935932537779, prime
173246558994 -> 234319118171, prime
173246558995 -> 5178319433153, prime
173246558996 -> 276187377107, prime
173246558997 -> 357748852999, prime
173246558998 -> 2248213489919, prime
173246558999 -> 173246558999, prime
173246559000 -> 235475397239, prime
173246559001 -> 2311636476749, prime
173246559002 -> 21136432161637 = 13*325921*4988569

***

Giorgos wrote:

Here is a set of 12 consecutive integers

 

173246558990->{2^1,5^1,241^1,71886539^1}->2524171886539->True
173246558991->{3^1,11^1,25033^1,209719^1}->31125033209719->True
173246558992->{2^4,13^2,43^1,1490011^1}->213431490011->True
173246558993->{19^1,3593^1,2537779^1}->1935932537779->True
173246558994->{2^1,3^3,431^1,911^1,8171^1}->234319118171->True
173246558995->{5^1,1783^1,19433153^1}->5178319433153->True
173246558996->{2^2,7^1,6187377107^1}->276187377107->True
173246558997->{3^1,57748852999^1}->357748852999->True
173246558998->{2^1,24821^1,3489919^1}->2248213489919->True
173246558999->{173246558999^1}->173246558999->True
173246559000->{2^3,3^1,5^3,47^1,53^1,97^1,239^1}->235475397239->True
173246559001->{23^1,1163^1,6476749^1}->2311636476749->True
173246559002->{2^1,11^1,3643^1,2161637^1}->21136432161637->False

 

***

Jean-Marc wrote:

Set of consecutive integers that generates 12 primes beginning with 1034119577213

1034119577213 -> (1034119577213) and 1034119577213 is prime.
1034119577214 -> (2 * 3^3 * 29 * 5189 * 127261) and 23295189127261 is prime.
1034119577215 -> (5 * 62773 * 3294791) and 5627733294791 is prime.
1034119577216 -> (2^7 * 11 * 2381 * 308467) and 2112381308467 is prime.
1034119577217 -> (3 * 49177 * 7009507) and 3491777009507 is prime.
1034119577218 -> (2 * 7 * 13 * 179 * 31742881) and 271317931742881 is prime.
1034119577219 -> (1034119577219) and 1034119577219 is prime.
1034119577220 -> (2^2 * 3 * 5 * 17235326287) and 23517235326287 is prime.
1034119577221 -> (1034119577221) and 1034119577221 is prime.
1034119577222 -> (2 * 17 * 107 * 284254969) and 217107284254969 is prime.
1034119577223 -> (3^2 * 114902175247) and 3114902175247 is prime.
1034119577224 -> (2^3 * 269 * 480538837) and 2269480538837 is prime.
1034119577225 -> (5^2 * 7 * 5909254727) and 575909254727 is not  prime.
 

 

***