Free 100 Essential Things You Didn't Know You Didn't Know by John D. Barrow

it. This is not impossible in principle, but it is impossible in practice in a sensible period of time. A criminal with the world’s fastest computer at his disposal might crack the encryption in several years, but by then the codes and account numbers would have been changed.
    For this reason very large prime numbers are very valuable things, and some have been patented when written in a certain form. There is no limit to the number of prime numbers – they go on forever – but there is a largest one that we have been able to check for primeness by ensuring that it has no factors. There is no magic formula that can generate all prime numbers, and it is suspected that no such formula exists. If it did, and it was found, there would be a major crisis in the world. Any government agency that found it would undoubtedly keep it top secret. Any academic who found it and made it public without warning would bring the world tumbling down. All military, diplomatic and banking codes would become easily breakable by fast computers overnight. The world of online commerce would face a serious threat to its continued existence. We would have to move to iris, or fingerprint, or DNA-based recognition systems that relied on unique features of our biochemistry rather than numbers stored in our memories. But these new indicators would still need to be stored in a secure way.
    The factoring of prime numbers is a ‘hard’ problem. Even if it is cracked and shown to be an ‘easy’ problem by means of a magic formula, you might think that we could just use some other ‘hard’ problem to encrypt sensitive information so that it still takes ages to reverse the operation and extract it. However, it is known that if one of the problems we believe to be ‘hard’ could be shown to be ‘easy’ by means of some new discovery, then that discovery could be used to turn all the other computationally ‘hard’ problems into ‘easy’ ones. It really would be a magic bullet.
    fn1 The answer is 5569 + 389965026814369.

28
    Is This a Record?
    I always thought that record would stand until it was broken.
    Yogi Berra
    Records – remember those those little discs of black vinyl that your parents owned that made a noise when spun on a turntable at high speed? Well, mathematicians are more interested in the other sort of records: the biggest, the smallest and the hottest. Are they predictable in any way?
    At first you might think not. True, they follow a trend of getting ‘better’ – they wouldn’t be records if they didn’t – but how could you predict whether a Michael Johnson or an Ian Thorpe was going to come along and break record after record? Amazingly, the women’s world record in the pole vault was broken on eight separate occasions by Yelena Isinbayeva in one year alone. Records like this are not random in a very important sense. Each new record is the result of a competitive effort that is not independent of all the previous attempts at the same feat. Pole vaulters learn new points of technique and continually train to improve their weaknesses and refine their technique. All you can predict about records like this is that they will be set again, eventually, although there might be a very long wait for the next one.
    However, there are different sorts of records that arise in sequences of events that are assumed to be independent of each other. Good examples are record monthly rainfalls, record high or low temperatures in one place over hundreds of years, or the heights of the record highest tides. The assumption that each event is independent of its predecessors is a very powerful one that allows us to make a striking prediction about how likely it is that there will be a record – irrespective of what the record is for. It could be rain, snow, fall of leaves, water levels, wind speeds or temperature.
    Let’s pick on the annual rainfall in the UK as our example. In the first year that we keep records the rainfall must be a record.