|
Problems & Puzzles: Puzzles
From July 12-18, 2025, contributions came from JM Rebert. Simon Cavegn, Oscar Volpatti Rebert wrote:
I found:
k primes -> digit sum (both consecutive primes)
1 [2] -> digit sum: [2] (both consecutive primes)
2 [2, 3] -> digit sum: [2, 3] (both consecutive primes)
3 [2, 3, 5] -> digit sum: [2, 3, 5 ] (both consecutive primes)
4 [2, 3, 5, 7] -> digit sum: [2, 3, 5, 7] (both consecutive primes)
5 [402131, 402133, 402137, 402139, 402197] -> digit sum: [11, 13, 17,
19, 23] (both consecutive primes)
6 [6340271501, 6340271521, 6340271527, 6340271657, 6340271677,
6340271699] -> digit sum: [29, 31, 37, 41, 43, 47] (both consecutive
primes)
*** Simon wrote: k=1: 2 k=2: 2 k=3: 2 k=4: 2 k=5: 402131 k=6: 6340271501 k=7: Not found. Searched from 2*10^14 to 4.49*10^14 so far. *** Oscar wrote:
The set of one-digit primes {2,3,5,7} is a trivial solution for the case
k=4.
Considering only multi-digit primes and sets which can't be further
extended (neither forward nor backward).
Smallest set for k <= 6, starting at prime p:
k p
1 11
2 137
3 41
4 191
5 402131
6 6340271501 (Giovanni Resta)
I tried to identify some sets for k>=7, by:
picking a small prime s as candidate sum of digits;
enumerating all primes p whose sum of digits is s, in ascending order;
checking the digit sums of primes near p and identifying the unique set
to which p belongs.
So far, I found six sets with size k=7.
s= 5, d= 112, p=
100000000000000000000000000000
s= 7, d= 46, p=
200010000000000000010000000000
s= 11, d= 24, p=
101100010001001200110001;
s= 13, d= 30, p=
100000000022001100010011001101
s= 17, d= 23, p=
10020021040101110010101;
s= 19, d= 21, p=
201033000620001000001.
Probably none of them is the desired smallest set.
*** |
|||
|
|||
|
