Problems & Puzzles:
Primes P=A&B such that...
Here we ask for primes P
that you divide exhaustively in two parts A&B such that:
- abs(A-B) is prime, for all the partitions of P
This is my largest example: P=60000607 because all of these
Q1. Can you find a larger one example?
Q2. Redo Q1 but using (A+B) instead of abs(A-B).
Contributions came from Emmanuel Vantieghem, Giovanni Resta and Hakan
I could not find a bigger
example than yours for Abs(A-B). If there is one, it will be >
For the sum, I found much
more solutions. My biggest one is 608844043. If there exists
a bigger one, it will be > 2*10^16. In my oppinion, there are no
Question Q2 of puzzle 709 correspond to Puzzle 401 - Magnanimous
In that puzzle I searched for prime and non prime such numbers up to
digits. I extended the search to 16 digits without finding further
For question Q1 I searched up to 16 digits, and the largest prime I
found is 60000607, that you already found.
Extending the search to composite numbers which produce primes, the
largest example I found (up to 16 digits) is 537353996.
Just for curiosity, I have considered also this variation:
primes such that dividing them in two parts in all possible ways,
the floor of the division of the largest part by the smallest part
is always a prime.
Up to 10^14 the largest such prime is 5335039446259.
533503944625 / 9 = 59278216069
53350394462 / 59 = 904243973
5335039446 / 259 = 20598607
533503944 / 6259 = 85237
53350394 / 46259 = 1153
5335039 / 446259 = 11
533503 \ 9446259 = 17
53350 \ 39446259 = 177
5335 \ 039446259 = 7393
533 \ 5039446259 =
53 \ 35039446259 = 661121627
5 \ 335039446259 = 67007889251
and all the truncated ratios (59278216069, 904243973, etc.) are
My largest example: P=608844043