Simple Mathematical Questions I do not (yet) Understand
Professor of Pure Mathematics
University of Southampton
‘Mathematics has beauty and romance. It’s not a boring place to be, the mathematical world. It’s an exciting place; it’s worth spending time there’. Marcus du Sautoy
Mathematics has been a major force in understanding science, engineering and technology and providing philosophical insights into the world at large. It is, unusually, one of the few disciplines straddling the boundary of art and science. It can also show contrasting sides. On the one hand, there is a highly theoretical side (pure maths) and, on the other, useful practical aspects (applied maths). At times pure concepts, often pursued for their lack of worldly applicability, can suddenly assume a commercial or political imperative. Number theory, for example, for many years an arcane backwater of mathematics, became suddenly important in the 1930’s and 1940’s for codebreaking. More recently, hyperbolic geometry has developed a commercial applicability to internet communications.
Insight into, and interpretation of the natural world is adequately understood by mathematicians. However, like any discipline it is still a little woolly around the edges in certain areas. Jim Anderson, a mathematician from Southampton University, gave an in-depth, erudite and amusing presentation of aspects within mathematics where he, and mathematical colleagues, refreshingly demonstrated a certain ignorance. Some questions have been solved but are still difficult to understand. Others are work-in-progress and some of these (Millennium Prize Problems) are subject a significant financial reward from the Clay Mathematics Institute for the first correct solution. There were originally seven millennium prizes, now down to six because one problem, the Poincaré conjecture, is considered to have been solved.
As befits a language, mathematics has its own nomenclature, a working knowledge of which provides a suitable backdrop before diving into deeper concepts. So, a ‘conjecture’ is a conclusion formed on the basis of incomplete information, a ‘theorem’ is a general proposition not self-evident but evidenced by a chain of reasoning, a ‘proposition’ is a statement expressing an opinion and an axiom is a statement or principle that is generally accepted to be true but need not be so (e.g. Newton’s Laws of motion).
To confuse matters, Fermat’s Last Theorem (FLT), which was solved in 1994, was technically a conjecture before the correct solution popped up but has always been known as a theorem. As in any field, convention is sometimes broken. Perhaps this was because Pierre de Fermat had apparently written in the margin of a book (1637) that he had a solution to the conjecture (theorem), but lack of space precluded him including it. Nothing was found after his death in his papers to suggest a solution had been achieved. It may be that Fermat’s son mischievously added this comment after his father’s death.
To continue the primer, a perfect number a positive integer equal to the sum of its divisors (e.g. 6 = 3+2+1; 28 = 14+7+4+2+1). A prime number is one divisible by 1 and itself and cannot be formed by multiplying two other numbers together and a natural number is a positive integer (whole number).
B. The Questions
1) How many perfect numbers are there?
In ~300 BC, Greek mathematician, Euclid showed that if 2p-1 is prime, 2p-1(2p-1) is a perfect, even number and all even perfect numbers have this form. For (2p-1) to be prime, p itself needs to be a prime. The list of perfect numbers generated by this formula becomes exponentially larger very quickly (6, 28, 496, 8128, 33,550,336 etc). Interestingly, only the first four were known to Ancient Greek mathematicians. The ability to generate a list of perfect numbers leads naturally to the question, how many perfect numbers are there?
The expression (2p-1) is known as a Mersenne prime and its inclusion in Euclid’s equation for perfect numbers indicates that the search for perfect numbers is related to the search for Mersenne primes. Every Mersenne prime yields an even perfect number, and vice versa. To prove that there are infinitely many even perfect numbers is equivalent to proving that there are infinitely many Mersenne primes. There are only 36 known Mersenne primes and the continued search for them will require huge amounts of computer time. Jim does not expect an answer to this question within his lifetime.
2) Collatz conjecture
A Collatz sequence is a series of numbers relevant to the Collatz Conjecture which was first proposed in 1937 by Lothar Collatz. Starting with a natural number n, each term in the sequence is derived from the previous term as follows: if n is even, the next number is n/2; if odd, the succeeding number is 3n+1. Each term in the sequence is derived from the previous in the same manner.
e.g. For n= 9: 9,28,14,7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1.
For n= 8: 4,2,1
For n= 6: 6,3,10,5,16,8,4,2,1
For n= 5: 5,16,8,4,2,1.
An immediately notable feature of these Collatz sequences is that they all conclude in 1. The conjecture asks whether, whatever number is selected for n, the sequence will always reduce to 1. Prominent Hungarian mathematician, Paul Erdős, described the Collatz Conjecture is one of the most difficult problems in maths today and is purported to have prophetically exclaimed that ‘Mathematics is not ready for such a problem’. Despite its difficulty, Erdős offered a prize of only US$500 for a correct solution which Jim notes in no way represents the sheer amount of time and effort which would be needed to solve this problem.
3) Four colour theorem
This theorem originated from the simple act of someone nonchalantly colouring a map of UK counties and wondering how few colours are needed without adjacent counties having the same colour (see Fig.1). This seemingly straightforward problem reached Augustus de Morgan, a professor of Mathematics at UCL, who presented it as a conjecture. It is a difficult proof, requiring all maps to demonstrate only 4 colours are required, or to find one map needing 5 colours. In 1879, the theorem was considered proved, only for errors to be found in the original proof ten years later, thereby downgrading the theorem back to a humble conjecture.
This proof was completed by Kenneth Appel and Wolfgang Haken at Illinois University in 1976. The key lies, counterintuitively, in being able to simplifying all maps into a series of networks. These consist of individual, building block components (of which there are several hundred). Demonstrating that each network component can be represented by no more than four colours proves the conjecture. The several hundred different network components and all were bundled through a computer checking procedure.
This somewhat massive checking of cases, whilst perfectly valid, was initially looked down on by the mathematical establishment because it represented a brute force attack on the conjecture and lacked any elegance. However, for anyone considering a career in cartography, five colours may be necessary where two separate regions belong to the same country and the map requires those regions to be the same colour.
Fig. 1. County map delineated by four colours, without a county of one colour lying next to a county of the same colour
4) Keio triangles of Hirawaka and Matsumgra
In 2018, Yoshinosuke Hirakawa and Hideki Matsumura of the Keio Institute of Pure and Applied Science (KiPAS) proved a conjecture, with a long-standing pedigree back to Ancient Greece, thereby presenting the world with a new theorem (Hirakawa-Matsumura theorem). This demonstrated that there is only one pair of triangles (an isosceles and a right-angled triangles) which have the same length of perimeter and the same area (see Fig.2). They used diophantine geometry and modern mathematical techniques to solve a question that had remained unanswered ~2,500 years.
Fig. 2 A Unique Pair of Triangles, proved using abstract modern Mathematics
5) Does P=NP?
Rather than being a straightforward algebraic expression (as it perhaps appears), it is the most important open question in computer science (see Fig.3).
This expression (formally defined in 1971) is one of the seven Millennium Prize Problems (now six, see Introduction) selected by the Clay Mathematics Institute with a US$1million prize for the first correct solution.
The equation asks whether (in polynomial time) a solution to a problem is easily solvable (P) and easily checkable for correctness (NP). Some challenges (see Sudoku- below) cannot be answered quickly, but if a solution is available, can be quickly verified (NP). In other words, P=NP asks, where a solution can be easily verified (NP) can it also be solved quickly (P)?
In Sudoku, for example, a player attempts to complete the grid by certain rules and any proposed solution can be easily verified by counting the numbers horizontally and vertically. In this case, Sudoku is quickly checkable (NP) but not quickly solvable (P). In contrast, standard maths operations such as +,-,x & ÷ are P problems and easily verified (NP). More complex problems such as scheduling, are, like Sudoku, difficult to solve, with its many possibilities of route choice etc, but easily verified, and are NP examples.
Fig. 3 Diagrammatic representation of P=NP and P≠NP. Any problem which is solved easily can be checked easily, but can any problem which is checked easily also be solved easily (i.e. are all NP problems really P?)
A proof either way (i.e. that P=NP or P≠NP) would have profound implications for mathematics generally, notably cryptography, Artificial Intelligence and game theory to name but a few. In cryptography, coded messages are hard to break but easy to verify. A proof that P=NP suggests that procedures or processes (algorithms) are available, but yet undiscovered, which would speed code-breaking, shattering the security of codes and ciphers. The converse (P≠NP) shows cryptography is still relatively secure and that generally there are problems which no amount of data could adequately solve.
6) Beal’s Conjecture
Beal’s conjecture was formulated in 1993 by Andrew Beal, banker, amateur mathematician, number enthusiast and poker player when investigating generalisations of Fermat’s Last Theorem (FLT). This shows that no three positive integers A, B and C satisfy the equation An + Bn = Cn for any value of n >2 and was brilliantly solved in 1995 by Andrew Wiles.
Beal’s Conjecture is a more general statement of FLT. The conjecture suggests that for Ax +By = CZ (where A, B, C, x, y & z are positive integers), and where x, y, z >2 then A, B and C must have common prime factors. The key to Beal’s Conjecture is to solve the equation (to prove the conjecture) or alternatively, to find counter example(s)(to disprove the conjecture). A prize of US$1million is available for the first correct solution, putting this on the same level as the Millennium Prize Problems.
Jim’s talk prompted many questions. In the language of mathematics, some were incorrect conjectures which Jim easily batted away, others inadequate propositions that were summarily dealt with; a few were viable suppositions which generated focussed discussion and for some, long held axioms were shown to be in need of revision. An excellent, complex, well-explained presentation.