A popular phrase used by skeptics is that "you can not prove a negative". The idea behind this phrase is that in the eyes of a believer in pseudo scientific concepts, the paranormal, cryptobiology or the like, it is often impossible to provide enough evidence of the absence of something to convince the believer beyond all doubt that "the something" does not exist. The phrase infers:-
"you can not prove a negative proposition beyond all doubt to a true believer".
A negative proposition is saying that something either does not exist or that something has no effect. Examples would be "extraterrestrial aliens are not visiting the Earth", "the Loch Ness Monster does not exist", "homeopathy does not work", "the Earth is not flat". A pedant might say that there is no clear difference between a negative and a positive proposition, for instance all of the above examples are arguments supporting the overwhelming lack of evidence and so could be looked at as positive propositions. "The Earth is not flat" is a statement supporting the overwhelming evidence that the Earth is a spheroid. It could however also be a statement supporting a notion that "the Earth is a cube". Intuitively though, the average person will be able to determine the difference between a negative and a positive proposition, "this formula works" versus "this formula does not work".
In the context of the statement "you can not prove a negative" we are not inferring that you can't prove a negative equation and we are not concerned with observations that if you throw a double negative in front of a positive you can make any statement into a negative. For example, "the Earth is a spheroid" versus "the Earth is not not a spheroid". Simply expressing something the second way does not automatically make it a negative proposition. It is still a positive proposition because two negatives always cancel each other out.
Having established what we are actually referring to when we say "you can not prove a negative", I will now refer you to an article by Steven D. Hales in which he discussed this very issue. However I would caution that Professor Hales takes a little time to get to the real meaning of the phrase. You can virtually skip to the second page to get to the part addressing the nature of the claim.
Professor Hales says "you can prove a negative beyond all reasonable doubt", which is exactly how reasonable people think. He makes a strong argument for inductive reasoning as evidence of absence, or proving a negative proposition, which is fine for the vast majority of "reasonable people". Whilst Professor Hales may not have understood the intent of the "can not prove a negative" phrase, his final paragraph, in trying to explain why reasonable people can't dismiss inductive reasoning, actually supports the intended nature of the original claim. The claim "you can not prove a negative" is aimed at unreasonable people, the true believers of any claim that reasonable people would consider to be extremely fanciful. So in the intended nature of the claim it is correct, you can not prove a negative. Evidence of absence is not absolute proof, and absolute proof is often the only thing that will convince a true believer. In other words, you can not convince a true believer. You can not prove a negative.