Finding numbers whose product is a particular number?

Question

Is there a standard way to formulate and evaluate the following? Basically, I want to find 100 possibly distinct (some numbers can be repeated) real numbers ($0 < 1 + \frac{n_i}{100} < 5$) such that:

$\sqrt[100]{\prod_{i=1}^{100} \left ( 1+\frac{n_i}{100} \right )} = 1.07$

It is ok if there are multiple solutions.

EDIT: Apologies! I made a mistake in writing the constraint.

I believe this can be simplified as:

$\prod_{i=1}^{100} m_i = 1.07^{100}$

where $m_i = 1 + \frac{n_i}{100}$ and $0 < m_i < 5$

After this I'm stuck - Are there any numerical methods to solve this?

Answer 1

Suggestion:

1. Start with Karolis Juodelė's suggestion: all $n_i=7$.
2. Keep all $n_i=7$ for $i>2$.
3. Change $n_1$ a little bit and (similar to the implicit function theorem) solve the original equation for $n_2$ as a function of $n_1$.
4. Since you are moving $n_1$ just a bit, $n_2$ will move just a bit as well, so all inequality constraints will hold.
5. After you got new values of $n_1$ and $n_2$ then you can leave their values fixed, and move on to the next numbers, $n_3$ and $n_4$ and so on...