Odd perfect numbers, Diophantine equations, and upper bounds

We obtain a new upper bound for odd multiperfect numbers. If $N$ is an odd perfect number with $k$ distinct prime divisors and $P$ is its largest prime divisor, we find as a corollary that $10^{12}P^{2}N<2^{4^{k}}$. Using this new bound, and extensive computations, we derive the inequality $k\geq 10$.