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Discover Ludwig"should write down" is correct and usable in written English.
It usually implies that it is a good idea to record what is being said or discussed, such as in this example sentence: I think you should write down the instructions for assembling the bookcase so you can easily reference them later.
Exact(60)
They should write down everything they remember.
"Here's what you should write down," he instructed the reporter.
Students should write down their opinions and then discuss the pros and cons.
Students should write down, as accurately as possible, their questions and the answers they receive.
Students should write down the answers to these questions with respect to their song choice.
On one side, they should write down what a typical newspaper reader learned on each day of the crisis.
Those rules concern when, and how, banks should write down the value of loans on their books.
True, Bennett told him he was funny and should write down what he says, but nothing more.
After settling on their "one-kilogram standard," each pair should write down their "recipe" for the standard.
Then groups should rephrase the modifications to the steps to specify their "secrets," which they should write down on a slip of paper.
The Federal Reserve's chairman, Ben S. Bernanke, has said mortgage companies should write down the amount borrowers owe in some cases.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com