Problems & Puzzles: Puzzles

Puzzle 116. A=B+C, A*B*C is a primorial...

Naohiro Nomoto asks for numbers A, B & C such that the following four rules apply:

1. A = B + C
2. A*B*C is a primorial
3. A, B, & C are, each, a product of n of distinct primes
4. gcd(A,B)=gcd(A,C)=1

Nomoto has found only two examples:

5 = 2 + 3

1105 = 231 + 874 [ 5*13*17 = 3*7*11 + 2*19*23 ]

Question 1: Find the next example

If the condition 3 is relaxed and we let that A, B & C are the product of distinct primes, not necessarily the same quantity, example:

10 = 3 + 7  [ 2*5 = 3 + 7 ]

There are some more examples, being the largest currently found, the following one:

496961=495726+1235 [ 17x23x31x41 = 2x3x7x11x29x37 + 5x13x19 ]

Question 2. Find the next 3 examples

Question 3. Can you argue if this sequence is finite or infinite?

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Notes:  All the numbers for the sequence described in the question 2 are a subset of the Sloane's integers sequence A057035.

Solution