Let $x=2000$, so the numerator is $$x^{5}+x^{4}+1=\left(x^{2}+x+1\right)\left(x^{3}-x+1\right)$$ (This latter factorization can be noted by the fact that plugging in $\omega$ or $\omega^{2}$ into $x^{5}+x^{4}+1$ gives 0 .) Then $x^{2}+x+1=4002001$ divides the numerator. However, it can easily by checked that 101 doesn't divide 4002001 (since, for example, $101 \nmid 1-20+0-4$ ), so 4002001 is one of the primes. Then the other one is $$\frac{2000^{3}-2000+1}{101} \approx \frac{2000^{3}}{101}>2000^{2} \approx 4002001$$ so 4002001 is the smaller of the primes.
The number $$316990099009901=\frac{32016000000000001}{101}$$ is the product of two distinct prime numbers. Compute the smaller of these two primes.