|
Problems & Puzzles: Puzzles
From July 26 to Aug. 2, 2025. contributions came from, Oscar Volpatti, *** Oscar wrote:
I wasn't able to find new approaches.
Exaustive search with fixed digit sum 23 found no solutions with 18
digits or less.
Previous work provides several constraints on desired primes p1...p7 and
digit sums s1...s7.
p7 has at most 21 digits, hence s7 is smaller than 21*9 = 189:
s7 is at most 181 and s1 is at least 23 (values up to 19 already
checked);
within such range, the cumulative gap s7-s1 is at least 24;
minimum value obtained for s1 = 23,29,37,59, or 89;
consequently, the cumulative gap p7-p1 must be at least 798;
minimum value obtained when:
either p1 ends with 001 and p7 ends with 799,
or p1 ends with 101 and p7 ends with 899,
or p1 ends with 201 and p7 ends with 999.
Among primes with 21 digits or less, the average gap is less than 50 and
the average cumulative gap is less than 300, so the value 798 is already
quite big.
I decided to check only primes matching such "minimum" gap.
Among primes with digit sum 29 and ending with 001, I found the
following solution with 19 digits:
s p
29 1212030150560200001
31 1212030150560200003
37 1212030150560200261
41 1212030150560200463
43 1212030150560200483
47 1212030150560200577
53 1212030150560200799
*** |
|||
|
|||
|
