Problems & Puzzles: Puzzles

Puzzle 1165  Two puzzles from Mr. N. Nomoto Mr. Naohiro Nomoto sent two puzzles. In this page I will publish only the first one. This is a problem about pairs of prime numbers. Definition:  P_{n}(x) = All products of pairs of odd prime numbers with a difference x below the nth prime number. e.g. For n = 4, x = 2, The fourth prime number is 7. Below 7, the pair of odd prime numbers with a difference of 2 is (3,5) and (5,7). Therefore, P_{4}(2) is 3*5*5*7. e.g. For n = 5, x = 2, The fifth prime number is 11. Below 11, the pair of odd prime numbers with a difference of 2 is (3,5) and (5,7). Therefore, P_{5}(2) is 3*5*5*7. The first puzzle: For n = 14, A002110 is the Sloane's sequence. When d chooses the 6,8,12,14,16,28,30,34,36,38, the product of P_{14}(d) is ( A002110(14)/2 )^6. Moreover, when k chooses the 2,4,10,18,20,22,24,26,32,40, the product of P_{14}(k) is ( A002110(14)/2 )^6. d and k are sets of even numbers that have no common elements. Question: Product of P_{n}(d) = Product of P_{n}(k) = ( A002110(n)/2 )^m　. Find d and k that satisfy this equation. d and k are sets of even numbers that have no common elements.

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During the week from 2 to 8 March 2024, contributions came from Alessandro Casini, JM Rebert,

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Alessandro wrote:

The number N must obviously be even, otherwise each prime would appear an odd number of times and it would be impossible to distribute them equally between the two products. There are no other solutions for n <= 30. The general problem of Equal Product partition is strongly NP-hard.

A few hours later he added:

Sorry, my bad. I was convinced that the puzzle was to find D and K as complete 2-partitions, i.e. they were disjoint and collected all possible even numbers for that N. I inserted an extra constraint due to a reading error. Having said that, in this case what I said remains and it's true.

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Rebert wrote:

I found:
 

n d k

7 [4, 8, 12] [2, 6, 10, 14]

8 [8, 10, 12] [2, 4, 6, 14, 16]

9 [2, 6, 16, 20] [4, 8, 10, 12, 14, 18]

11 [8, 16, 24] [2, 4, 10, 12, 18, 20, 26]

13 [6, 14, 16, 22, 26, 28, 30, 36] [2, 4, 8, 10, 12, 18, 20, 24, 32, 34, 38]

14 [6, 12, 16, 28, 32, 34, 38] [2, 4, 8, 10, 14, 18, 20, 22, 24, 26, 30, 36, 40]

15 [6, 12, 16, 32, 34, 38, 44] [2, 4, 8, 10, 14, 18, 20, 22, 24, 26, 28, 30, 36, 40, 42]

17 [6, 8, 10, 12, 16, 26, 34, 36, 38, 52, 54, 56] [2, 4, 14, 18, 20, 22, 24, 28, 30, 32, 40, 42, 44, 46, 48, 50]

19 [14,16,54] [2,8,12,20,24,30,34,40,44,46,56,62]

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