Problems & Puzzles: Puzzles

 Puzzle 709 Primes P=A&B such that... Here we ask for primes P that you divide exhaustively in two parts A&B such that: P=A&B abs(A-B) is prime, for all the partitions of P This is my largest example: P=60000607 because all of these are primes abs(6-0000607) abs(60-000607) abs(600-00607) abs(6000-0607) abs(60000-607) abs(600006-07) abs(6000060-7) 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 Summakoglu

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Emmanuel wrote:

I could not find a bigger example than yours for  Abs(A-B).  If there is one, it will be > 2*10^15.
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 more sollutions.

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Giovanni wrote:

Question Q2 of puzzle 709 correspond to Puzzle 401 - Magnanimous primes.
In that puzzle I searched for prime and non prime such numbers up to 15
digits. I extended the search to 16 digits without finding further
terms.

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.
Indeed
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 = 9454871
53 \ 35039446259 = 661121627
5 \ 335039446259 = 67007889251

and all the truncated ratios (59278216069, 904243973, etc.) are primes.

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Hakan wrote:

Q2: My largest example: P=608844043

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