Ok, so we have now established what P and NP are, and how they are interdependent. What hasn’t yet been covered though, is why you should care. It might seem tricky to answer – why should a non-geek care? In fact it isn’t hard to answer at all. The safety of your money depends on P vs. NP.  The cost of computer-driven devices may also depend on P vs. NP. It’s even arguable that some people’s jobs even depend on P not equalling NP.

In terms of money safety, I am speaking of the encryption method called Public Key Cryptography. This is the method that online shops use so that you can transfer payment details without people being able to eavesdrop on you, and gain access to your money. Within a lot, if not all of the Public Key Encryption algorithms, a huge number is generated and used as part of a private key, which only you and your computer knows. They crux of this is that the private key often consists of large prime factors, and as it stands there is no prime factorisation method for current, non-quantum computers which is in P-time, so if you have a large enough number,  it is practically impossible to find the factors and hence your private key. If P was to equal NP, then prime factorisation would be in P-time, hence rendering Public Key Encryption weak, and making your payment transactions insecure.

A fascinating result of P equalling NP would be that the Travelling Salesperson Problem would be in P-time. This problem deals with route-finding, and at present, it has no P-time solution, making it practically unusable for routes which include a lot of stopping points. The obvious applications of this would be logistics, as being able to quickly plan a tour of 500 stops based on fuel used and distance, among other factors, could save a lot of money compared to current methods, as well as reducing environmental impact through cutting back on wasted mileage. Surprisingly though, this problem, if solved in P-time would make the production of microchips a lot more efficient. This would be due to the routing of the wires between components – there are many components in single microchips, and wiring them up is likely to involve a lot of wastage, which is passed on to the customer through extra cost. By applying Travelling Salesperson to the designs of the chip, the wiring could be much more efficient, saving in production costs and hence saving the consumer money. As microchips are used in a huge amount of products now, from computers to air-fresheners, this will allow the consumer to save a lot of money on various products.

This brings me to my final point, which is a more worrying one. Should P equal NP, then a lot of jobs done by humans as computers could not do them efficiently might disappear. Business is ruthless, and the unfortunate reality is that if a computer can do it, the human is no longer needed. Jobs which might be affected by these developments would be route-planning, some clerical work, among many different fields where this would hit. For this reason, a lot of people working in these fields will probably be praying for P not to equal NP, if they know about it at all.

That brings me to the end of this computing journey, and although a lot of words have been written, I have only covered one small part of Computer Science as a whole. Hopefully you will have seen from this that P vs. NP has real world consequences, and solving it is a very big deal – there is a $1 million reward for the person who submits a correct solution to it, from the Clay Mathematics Institute as part of the Millennium Prize Problem list. Personally, I don’t believe P equals NP. I think that by now, a problem in NP-Complete would have been found that is actually a P-time algorithm, and we would have been done with it. As it stands, there are over 3000 NP-Complete problems, and none have yet been found to be in P-time. As for the actual result? Well, we’ll have to wait and see. Let’s just hope we don’t end up with another Fermat’s Last Theorem.