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“Definitive?”
When you have plenty of time to devote to a skill’s task and that task has no adverse effect upon failure, the GM might rule that you can take 20 on that skill check. This is similar to taking 10, but instead of assuming your roll was a 10, you assume it’s a 20.
There are no listed negative effects for failing a Gather Information check, so if a GM wants to allow you to take 1d4x20 hours, they can. Or they can say no.

LankyOgre |
If you want to take several days to ferret out the information, and if your 20+Mod would beat the DC, then go for it. Keep in mind, depending on the type of information you are asking about, you could tip your hand and reveal your plans to a rival or subject of investigation. Also, Many situations have their own built in time restrictions that a few days may ruin.

Metaphysician |
I would generally allow it, but with the proviso that they would definitely *also* earn the "negative downside for failure", which is to say, anyone interested will *know* they are looking for information. Yes, you normally can't take 20 if there is a failure effect, but in this case the effect is only loosely connected to the effort in terms of time, so it still works.
Note that this can also serve as a useful benchmark for how much information someone is guaranteed to eventually turn up, especially if that someone is an authority that doesn't have to worry about being noticed. Say the PCs do some kind of action that comes to the attention of the local police, enough to put diligent and professional effort into investigating it. Figure out the relevant skill bonus for whatever detectives are looking into it. After a couple hours, assume a Take 10 with that bonus, compare it to a DC based on the efforts or lack thereof of the PCs to cover their tracks or avoid attention. After a week, assume a Take 20. Thus you can determine how soon any police, or APBs, show up on the PCs' doorstep. If they do a good enough job the whole thing might be a dead end, but effort to hide their tracks could still be the difference between having a few hours dead start, versus a full week.