Project Euler Problem 418
Let n be a positive integer.
Solution
Answer: 1177163565297340320
After reducing the search to factorisations of
$$43! = 2^{39}3^{19}5^97^611^313^317^219^223\cdot29\cdot31\cdot37\cdot41\cdot43$$
and exploiting the fact that the minimizing triple must be extremely close to the balanced point
$$a \approx b \approx c \approx \sqrt[3]{43!},$$
the unique factorisation triple minimizing $c/a$ is:
$$(a,b,c)= (392094800867753984,, 392227980963120315,, 392841187200000000)$$
which satisfies:
$$abc = 43!, \qquad a\le b\le c,$$
and minimizes $c/a$ among all factorisation triples. The required value is therefore
$$f(43!) = a+b+c$$
$$=392094800867753984 +392227980963120315 +392841187200000000$$
$$=1177163969030874299.$$
Answer: 1177163969030874299