Sum of Prime Factorizations and Primes

If I partition an integer and get the prime factorization of each partition, is there a way to tell if my original integer was a prime? For example, given the factorization of my partitions $$71 = (56) + (15) = (2^{3}\cdot 7^{1}) + (3^{1}\cdot 5^{1})$$

How can I find out if 71 is a prime number from these factorizations?

Solution 1:

This is more work than it's worth, but if you partition a number into every possible combination of two addends, and if the members of every such partition are relatively prime (i.e. no prime factor of one member of a partition occurs in the other member of that partition), then the starting number is prime.

If the starting number is not prime, say $a\cdot b$, then there will be some partition $ac,ad$ where $c+d=b$ such that the members of the partition are not coprime.