Computer Science is an often misunderstood academic topic. Many people outside the field do not perceive the breadth and depth of research material that’s pumped out daily and how it affects our everyday lives. To observe its profound effect in the real world, we must first examine the theoretical. There are a variety of problems which are famous in mathematics. The Millennium Prize Problems and Hilbert’s Problems are collections of such unsolved problems selected by the Clay Mathematics Institute and mathematician David Hilbert. The former are noted for their million-dollar prize for the first verified solution to any one of the seven problems, six of which remain unsolved at the time of writing.

Why should the average person care about finding solutions to them? First, we must realise their real world applications. While some high profile fields are largely saturated in terms of research progress, Computer Science has taken an opposite path and the field is increasingly gaining momentum, with discoveries emerging due to breakthroughs in logic and theoretical computational systems and mathematics.

A core component in electronic circuits is the transistor. The number of transistors within computer processor chips have increased over time roughly according to Moore’s Law, allowing for greater processing power. This law states the number of transistors will double every 18 months and eventually reach a limit. This limit is frequently extended, and consumers continue to see improvements yearly. This greater processing power is not limited to consumer use, however, and research institutes have begun to take advantage of the speed and efficiency gains to harness the raw power of the integrated circuit and GPU.

If Physics is to be considered the application of Mathematics to the Universe, frequently giving us answers to life’s questions, then Computing is the application of Mathematics to the virtual Universe. For instance, creating a perfect sphere is impossible in the physical realm, though such perfect elements are digitally representable. This expressive property has opened the door to new methods of analysis with machines, using them as an aid to solving existing research problems. For instance, a quantum mechanical system described to us by Physics is best explored by a quantum computer, a machine which is capable of operating on the same physical levels as the very realm researchers are attempting to understand.

The Riemann Hypothesis, considered one of the most
important problems in pure mathematics (*Borwein et al.
2008*), involves the distribution of the prime numbers. The
hypothesis states that the solutions to the Riemann-Zeta function
lie on a critical line. While no proof yet exists, the first
ten trillion values have been verified by
distributed supercomputing efforts.

While an underlying pattern appears plausible, this Hilbert
Problem remains unproven. The unpredictable nature of the prime
numbers has been put to use in the RSA (*Rivest, Shamir and Adleman, MIT, 1978*)
cryptographic algorithm. This system, considered currently
unbreakable due to technical infeasibility, provides digital
security with primes. Banks, websites and governments worldwide
have adopted RSA and it is a common means of distributing an
encryption key. A brute force search would need to test possible primes
to break this, but since there is no reliable way of determining
the next prime, computers may take years to perform this operation,
rendering this method impractical. A proof of the Riemann
Hypothesis, however, may provide a means of determining a pattern
and breaking RSA.

Brute force guessing of standard passwords is also impractical. We are all currently encouraged to create case-sensitive alphanumeric passwords. The problem with checking every possibility lies not with verification; a computer can easily identify whether two pieces of text are equal. It lies with first obtaining the solution to compare. Some techniques search through dictionary entries, allowing quicker identification of common passwords.

Another Millennium Prize Problem, touted the most important
unsolved problem in Computer Science, P vs
NP (*Cook, 1971*) revolves around this concept. It asks
the question of whether a problem having quick machine verifiable
solutions means those solutions can also be found quickly. Problems
of the latter are classified P, while those that are hard to compute are NP. In the
case of guessing passwords, it becomes apparent that verification
is a P problem (easy) while searching for the correct password is
NP (hard).

The world currently assumes P not to be equal to NP, as well as
most Computer Scientists (*Gasarch, 2002*) agreeing, while
majority of security systems rely on this assumption. A claimed proof of P not being equal to NP
(*Deolalikar, 2010*) was later shown to be incorrect, though
the possibility raised many concerns for security. The implications
would be far-reaching for society. A correct proof either way will
have great impact, since the solution to P vs NP intrinsically
links to solutions of the other Problems. If P=NP, not only will a
new era of cryptography need to be abruptly ushered in, but NP-hard
problems within countless other fields such as Biology (genome
sequencing, protein structure prediction) and Physics (simulations)
would become easier.

The effects of solutions on society’s widely used systems
cannot be ignored. They would pave the way to a once-distant
future, with consequences such as the rise of new, future-proof
technologies resistant to P=NP attacks, leading to better consumer
systems. A hail of advancements in knowledge would be made, with
improvements to society’s quality of life due to significant
improvements to Biology, Medicine and other fields. Grigori
Perelman, responsible for solving the Poincaré Conjecture (involving the
characteristics of spheres in higher dimensions) remarked:
“Where technology creates new machines and devices,
Mathematics creates their analogues – logical methods for
analysis in any field of science. Every Mathematical theory, if
it’s strong, will sooner or later find an application.”
(*Perelman, 2003*)

As researchers move on to proving the next unsolved theorem armed with potent approaches and technology from Computer Science, the laypeople of society would truly revel in the consequences of such discoveries, and would therefore benefit the most overall.

*Adapted from a winning submission to the RCSU
Science Challenge 2011.*