Meaning of "at minimum"?

[hereinchina]

Hello,
Do the expressions "at minimum" and "at the minimum" have the same meanings as the expressions "at the least" and "at the very least"? My understanding of the meaning of the expressions "at the least" and "at the very least" is that they mean "not less than and probably much more than." Also, am I using the expressions correctly in the following sentences?
1. This painting costs one million dollars "at the least".
2. This painting costs one million dollars "at minimum".
3. This painting costs one million dollars "at the minimum".
4. "At minimum", you should have warned your neighbors the building was on fire before you ran out.
5. "At the minimum", you should have warned your neighbors the building was on fire before you ran out.
6. "At the least", you should have warned your neighbors the building was on fire before you ran out.

[fluffyhamster]

I haven't heard of 'at minimum', so I'd rather completely discount examples 2 and 4. And even if there is evidence that others use it, I don't think I'll be using it myself anytime soon!

[ouyang]

I would advise my students to use "at the very least" rather than "at minimum" or "at the minimum", but I think they're all fairly common expressions in America. I would also encourage "The least you should have done was warn your neighbors."

I think "costs at least one million dollars" is much better than any of your examples. Saying something like "At the very minimum, it would cost one million dollars" implies that the listener has significantly underestimated the cost.

[ronqgentry]

I think Ouyang is more than right. All of these examples are grammatically correct and may be heard in the United States, even "at minimum". It is just a question of which is most popular. Another thing to consider is the situation in which you are using the phrase. In an academic situation "minimum" may give the impression of more education or in a financial situation "minimum" may give a feeling of more expertise. This is probably because "minimum" is a Latin root word and "least" is an old English Anglo-Saxon word. We frequently find words like this in English where we give more importance to Latin root words or a stronger meaning as in Ouyang's other example: that at minimum means greatly underestimated. Any other opinions out there?

The difference in "at minimum" and "at the minimum" is just the article "the". This one of those cases where it is OK to drop the article in English. It's just less popular to say that way so you don't hear it as much.