"...imagine you wake up one morning with your head full of a complete proof of the Riemann hypothesis. (This is arguably the greatest open problem in mathematics, and is a deep statement about the distribution of the prime numbers, the atoms of arithmetic.) Giddy with anticipation of certain fame, you leap out of bed and type up the proof, in its full 500-page glory. That done, you are seized by doubt. How wise is it to inflict the full brunt of your genius on your colleagues? Will anyone even listen, or be prepared to check your proof?

What you can do is re-format your write-up into a (somewhat longer) PCP proof. Anyone now wishing to verify your argument need only pick a handful of words at random, and follow a set list of instructions to conclude whether it is correct or not. An error might have slipped in on account of faulty working, buggy reformatting, or outright cheating. No matter, it will be caught with overwhelming probability. What the reformatting step does is smear any error all over the proof, making it easier to spot. In much the same way, a diligent sandwich maker will smear a smidgen of jam evenly over his bread, rather than leaving it concentrated in one corner, and so make the whole more savoury.

...In its full splendour, PCP asserts that any statement S whose validity can be ascertained by a proof P written over n bits also admits an alternative proof, Q. This proof Q has two appealing features: it can be derived from P in a number of steps proportional to n^c, where c is some constant; and P can be verified by examining only three bits of Q picked at random. If S is true, a correct P will satisfy the verifier with a probability of 99%. If it is not true, any alleged proof P will trigger a rejection from Q with a probability higher than 50% (ref. 4). Not impressed with this error rate? Then all you have to do is pick, instead of three bits of Q, as many bits as are contained in this line of text. The error probability will drop to one in a billion."