Cube Numbers 1 x 1 x 1 = 1 1 is a cube number
2 x 2 x 2 = 8 8 is a cube number

5 x 5 x 5 = 125 125 is a cube number
6 x 6 x 6 = 216 216 is a cube number
7 x 7 x 7 = 343 343 is a cube number
8 x 8 x 8 = 512 515 is a cube number
9 x 9 x 9 = 729 729 is a cube number

10 x 10 x 10 = 1000 1000 is a cube number



3 x 3 x 3 = 27 27 is a cube number
4 x 4 x 4 = 64 64 is a cube number

Odd Numbers

. :. ::. :::. ::::. :::::. ::::::. and so on
1 3 5 7 9 11 13 and so on

ALL numbers NOT in the 2 times table are odd numbers
ALL odd numbers leave a remainder of 1 if divided by 2
ALL odd numbers end in 1, 3, 5. 7 or 9
Examples
We can tell that 71 is odd because it ends in 1
We can tell that 123 is odd because it ends in 3
We can tell that 2345 is odd because it ends in 5
We can tell that 64747 is odd because it ends in 7
We can tell that 567989 is odd because it ends in 9

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Even Numbers

: :: ::: :::: ::::: :::::: ::::::: and so on
2 4 6 8 10 12 14 and so on

ALL numbers in the 2 times table are even numbers
ALL even numbers divide by 2 without leaving a remainder
ALL even numbers end in 0, 2, 4. 6 or 8
Examples
We can tell that 70 is even because it ends in 0
We can tell that 132 is even because it ends in 2
We can tell that 1234 is even because it ends in 4
We can tell that 64646 is even because it ends in 6
We can tell that 567898 is even because it ends in 8

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Digits -- Value By Place

The value of a digit depends upon the position of the digit in the number.

In the number 8 the digit 8 simply tells us that the number of units is 8.

In the number 27 the digit 7 tells us the number of units
whilst the digit 2 tells us the number of tens thus the value of the 2 is 20.

In the number 456 the digit 6 tells us the number of units,
the digit 5 tells us the number of tens thus the value of the 5 is 50
and the digit 4 tells us the number of hundreds thus the value of the 4 is 400

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Maths -- Digits

In maths a digit is a single figure number
The digits are 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 and 9

Single digit numbers 0 to 9
Double digit numbers 10 to 99
Triple digit numbers 100 to 999

Unjamming Traffic



Traffic jams often occur for seemingly no reason, especially when you are going somewhere in a hurry. It is a common occurrence on busy roads to be brought to a stand-still when there is no ostensible cause for the delay. Mathematicians from the Universities of Exeter, Bristol and Budapest have developed a model of traffic behaviour that explains how an unexpected event as simple as a car changing lanes, or a van braking suddenly, can bring traffic to a grinding halt kilometres behind the incident. They have recently published their work .Their model suggests that when reacting to an unexpected event, drivers may slow down to below a critical speed, which then forces the car behind it to slow down further still. Eventually, cars further back in the queue must stop. This produces a wave travelling backwards from the point of disturbance.

The modelling is based on bifurcation theory, which studies how and when mathematical problems change from having only one possible solution to having many. Parameter values at which this change occurs are known as bifurcation points. In the traffic example, the important parameter is the average headway between cars on the road: if this value is large, then small incidents do not cause the system to change significantly. However, if it is too small, the cars do not have enough time to react to an incident and a stop-and-go wave can develop throughout the traffic.

The model uses a circular road of length L, around which n cars travel. The group suggests that this could be interpreted as traffic on a circular road around a large city such as the M25 around London. The cars and drivers are assumed to be identical, the ith vehicle follows the (i+1)th vehicle and the nth car follows the 1st. The model uses a differential equation that relates the motion of the cars to the headway:
where is the position of car is the length of the track, and are the time derivatives of — velocity and acceleration — and is a known sensitivity factor. is the known optimal velocity function and depends on the headway between car and the one in front: . As the cars are travelling in a circle, .

The team considered solutions for car position and velocity with regard to the headway parameter, and identified parameter values at which bifurcations occurred. Some variation in traffic speed can be absorbed by the system to maintain smooth traffic flow, however if the average headway is too small, the system no longer has the single steady flow solution in which the velocity of the cars remains smooth and above zero, but two solutions: a steady flow solution and one in which the traffic is stop-start and vehicle velocities periodically drop to zero. This second solution produces a stop-and-go wave. Such a backward travelling wave can die out by itself, or get worse, ending up as a persistent stop-and-go wave that travels around the whole circle.

The group predicts this behaviour on busy highways with more than 15 cars per kilometre. Heavy traffic does not automatically lead to congestion, but the model suggests, as every driving instructor teaches, that drivers should give themselves enough headway to react to an unforeseen event so that they do not have to slow down too suddenly. The authors suggest that overhead gantries on freeways could display temporary and variable speed limits that, if followed by the traffic, would overcome jams and return the traffic to uniform flow. The MIDAS system installed on the M25 motorway around London is currently able to provide this information to drivers. The team now plans to expand on the model and incorporate cars fitted with new electronic devices that increase reaction time and so cut down on over-braking.

Fermat's last theorem



Pierre de Fermat died in 1665. Today we think of Fermat as a number theorist, in fact as perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was in fact a lawyer and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous article written as an appendix to a colleague's book.

There is a statue of Fermat and his muse in his home town of Toulouse:
(Click it to see a larger version)


Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat's letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel written as a marginal note in his father's copy of Diophantus's Arithmetica.

Fermat's Last Theorem states that

xn + yn = zn

has no non-zero integer solutions for x, y and z when n > 2. Fermat wrote

I have discovered a truly remarkable proof which this margin is too small to contain.

Fermat almost certainly wrote the marginal note around 1630, when he first studied Diophantus's Arithmetica. It may well be that Fermat realised that his remarkable proof was wrong, however, since all his other theorems were stated and restated in challenge problems that Fermat sent to other mathematicians. Although the special cases of n = 3 and n = 4 were issued as challenges (and Fermat did know how to prove these) the general theorem was never mentioned again by Fermat.

In fact in all the mathematical work left by Fermat there is only one proof. Fermat proves that the area of a right triangle cannot be a square. Clearly this means that a rational triangle cannot be a rational square. In symbols, there do not exist integers x, y, z with
x2 + y2 = z2 such that xy/2 is a square. From this it is easy to deduce the n = 4 case of Fermat's theorem.

It is worth noting that at this stage it remained to prove Fermat's Last Theorem for odd primes n only. For if there were integers x, y, z with xn + yn = zn then if n = pq,

(xq)p + (yq)p = (zq)p.

Euler wrote to Goldbach on 4 August 1753 claiming he had a proof of Fermat's Theorem when n = 3. However his proof in Algebra (1770) contains a fallacy and it is far from easy to give an alternative proof of the statement which has the fallacious proof. There is an indirect way of mending the whole proof using arguments which appear in other proofs of Euler so perhaps it is not too unreasonable to attribute the n = 3 case to Euler.

Euler's mistake is an interesting one, one which was to have a bearing on later developments. He needed to find cubes of the form

p2 + 3q2

and Euler shows that, for any a, b if we put

p = a3 - 9ab2, q = 3(a2b - b3) then
p2 + 3q2 = (a2 + 3b2)3.

This is true but he then tries to show that, if p2 + 3q2 is a cube then an a and b exist such that p and q are as above. His method is imaginative, calculating with numbers of the form a + b√-3. However numbers of this form do not behave in the same way as the integers, which Euler did not seem to appreciate.

The next major step forward was due to Sophie Germain. A special case says that if n and 2n + 1 are primes then xn + yn = zn implies that one of x, y, z is divisible by n. Hence Fermat's Last Theorem splits into two cases.

Case 1: None of x, y, z is divisible by n.
Case 2: One and only one of x, y, z is divisible by n.

Sophie Germain proved Case 1 of Fermat's Last Theorem for all n less than 100 and Legendre extended her methods to all numbers less than 197. At this stage Case 2 had not been proved for even n = 5 so it became clear that Case 2 was the one on which to concentrate. Now Case 2 for n = 5 itself splits into two. One of x, y, z is even and one is divisible by 5. Case 2(i) is when the number divisible by 5 is even; Case 2(ii) is when the even number and the one divisible by 5 are distinct.

Case 2(i) was proved by Dirichlet and presented to the Paris Académie des Sciences in July 1825. Legendre was able to prove Case 2(ii) and the complete proof for n = 5 was published in September 1825. In fact Dirichlet was able to complete his own proof of the n = 5 case with an argument for Case 2(ii) which was an extension of his own argument for Case 2(i).

In 1832 Dirichlet published a proof of Fermat's Last Theorem for n = 14. Of course he had been attempting to prove the n = 7 case but had proved a weaker result. The n = 7 case was finally solved by Lamé in 1839. It showed why Dirichlet had so much difficulty, for although Dirichlet's n = 14 proof used similar (but computationally much harder) arguments to the earlier cases, Lamé had to introduce some completely new methods. Lamé's proof is exceedingly hard and makes it look as though progress with Fermat's Last Theorem to larger n would be almost impossible without some radically new thinking.

The year 1847 is of major significance in the study of Fermat's Last Theorem. On 1 March of that year Lamé announced to the Paris Académie that he had proved Fermat's Last Theorem. He sketched a proof which involved factorizing xn + yn = zn into linear factors over the complex numbers. Lamé acknowledged that the idea was suggested to him by Liouville. However Liouville addressed the meeting after Lamé and suggested that the problem of this approach was that uniqueness of factorisation into primes was needed for these complex numbers and he doubted if it were true. Cauchy supported Lamé but, in rather typical fashion, pointed out that he had reported to the October 1847 meeting of the Académie an idea which he believed might prove Fermat's Last Theorem.

Much work was done in the following weeks in attempting to prove the uniqueness of factorization. Wantzel claimed to have proved it on 15 March but his argument

It is true for n = 2, n = 3 and n = 4 and one easily sees that the same argument applies for n > 4

was somewhat hopeful.
[Wantzel is correct about n = 2 (ordinary integers), n = 3 (the argument Euler got wrong) and n = 4 (which was proved by Gauss).]

On 24 May Liouville read a letter to the Académie which settled the arguments. The letter was from Kummer, enclosing an off-print of a 1844 paper which proved that uniqueness of factorization failed but could be 'recovered' by the introduction of ideal complex numbers which he had done in 1846. Kummer had used his new theory to find conditions under which a prime is regular and had proved Fermat's Last Theorem for regular primes. Kummer also said in his letter that he believed 37 failed his conditions.

By September 1847 Kummer sent to Dirichlet and the Berlin Academy a paper proving that a prime p is regular (and so Fermat's Last Theorem is true for that prime) if p does not divide the numerators of any of the Bernoulli numbers B2 , B4 , ..., Bp-3 . The Bernoulli number Bi is defined by

x/(ex - 1) = Bi xi /i!

Kummer shows that all primes up to 37 are regular but 37 is not regular as 37 divides the numerator of B32 .

The only primes less than 100 which are not regular are 37, 59 and 67. More powerful techniques were used to prove Fermat's Last Theorem for these numbers. This work was done and continued to larger numbers by Kummer, Mirimanoff, Wieferich, Furtwängler, Vandiver and others. Although it was expected that the number of regular primes would be infinite even this defied proof. In 1915 Jensen proved that the number of irregular primes is infinite.

Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved. It has the dubious distinction of being the theorem with the largest number of published false proofs. For example over 1000 false proofs were published between 1908 and 1912. The only positive progress seemed to be computing results which merely showed that any counter-example would be very large. Using techniques based on Kummer's work, Fermat's Last Theorem was proved true, with the help of computers, for n up to 4,000,000 by 1993.

In 1983 a major contribution was made by Gerd Faltings who proved that for every n > 2 there are at most a finite number of coprime integers x, y, z with xn + yn = zn. This was a major step but a proof that the finite number was 0 in all cases did not seem likely to follow by extending Faltings' arguments.

The final chapter in the story began in 1955, although at this stage the work was not thought of as connected with Fermat's Last Theorem. Yutaka Taniyama asked some questions about elliptic curves, i.e. curves of the form y2 = x3 + ax + b for constants a and b. Further work by Weil and Shimura produced a conjecture, now known as the Shimura-Taniyama-Weil Conjecture. In 1986 the connection was made between the Shimura-Taniyama- Weil Conjecture and Fermat's Last Theorem by Frey at Saarbrücken showing that Fermat's Last Theorem was far from being some unimportant curiosity in number theory but was in fact related to fundamental properties of space.

Further work by other mathematicians showed that a counter-example to Fermat's Last Theorem would provide a counter -example to the Shimura-Taniyama-Weil Conjecture. The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge, England the first on Monday 21 June, the second on Tuesday 22 June. In the final lecture on Wednesday 23 June 1993 at around 10.30 in the morning Wiles announced his proof of Fermat's Last Theorem as a corollary to his main results. Having written the theorem on the blackboard he said I will stop here and sat down. In fact Wiles had proved the Shimura-Taniyama-Weil Conjecture for a class of examples, including those necessary to prove Fermat's Last Theorem.

This, however, is not the end of the story. On 4 December 1993 Andrew Wiles made a statement in view of the speculation. He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However one problem remains and Wiles essentially withdrew his claim to have a proof. He states

The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semisquare case (of the symmetric square representation associated to a modular form) is not yet complete as it stands. I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures.

In March 1994 Faltings, writing in Scientific American, said

If it were easy, he would have solved it by now. Strictly speaking, it was not a proof when it was announced.

Weil, also in Scientific American, wrote

I believe he has had some good ideas in trying to construct the proof but the proof is not there. To some extent, proving Fermat's Theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest.

In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work. They tried a new approach with a similar lack of success. In August 1994 Wiles addressed the International Congress of Mathematicians but was no nearer to solving the difficulties.

Taylor suggested a last attempt to extend Flach's method in the way necessary and Wiles, although convinced it would not work, agreed mainly to enable him to convince Taylor that it could never work. Wiles worked on it for about two weeks, then suddenly inspiration struck.

In a flash I saw that the thing that stopped it [the extension of Flach's method] working was something that would make another method I had tried previously work.

On 6 October Wiles sent the new proof to three colleagues including Faltings. All liked the new proof which was essentially simpler than the earlier one. Faltings sent a simplification of part of the proof.

No proof of the complexity of this can easily be guaranteed to be correct, so a very small doubt will remain for some time. However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat's Last Theorem

Prime numbers



Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.

The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.
A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.
A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.
You can see more about these numbers in the History topics article Perfect numbers.

By the time Euclid's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.

Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. The mathematician Euler (much later in 1747) was able to show that all even perfect numbers are of this form. It is not known to this day whether there are any odd perfect numbers.

In about 200 BC the Greek Eratosthenes devised an algorithm for calculating primes called the Sieve of Eratosthenes.

There is then a long gap in the history of prime numbers during what is usually called the Dark Ages.

The next important developments were made by Fermat at the beginning of the 17th Century. He proved a speculation of Albert Girard that every prime number of the form 4 n + 1 can be written in a unique way as the sum of two squares and was able to show how any number could be written as a sum of four squares.
He devised a new method of factorising large numbers which he demonstrated by factorising the number 2027651281 = 44021 46061.
He proved what has come to be known as Fermat's Little Theorem (to distinguish it from his so-called Last Theorem).
This states that if p is a prime then for any integer a we have ap = a modulo p.
This proves one half of what has been called the Chinese hypothesis which dates from about 2000 years earlier, that an integer n is prime if and only if the number 2n - 2 is divisible by n. The other half of this is false, since, for example, 2341 - 2 is divisible by 341 even though 341 = 31 11 is composite. Fermat's Little Theorem is the basis for many other results in Number Theory and is the basis for methods of checking whether numbers are prime which are still in use on today's electronic computers.

Fermat corresponded with other mathematicians of his day and in particular with the monk Marin Mersenne. In one of his letters to Mersenne he conjectured that the numbers 2n + 1 were always prime if n is a power of 2. He had verified this for n = 1, 2, 4, 8 and 16 and he knew that if n were not a power of 2, the result failed. Numbers of this form are called Fermat numbers and it was not until more than 100 years later that Euler showed that the next case 232 + 1 = 4294967297 is divisible by 641 and so is not prime.

Number of the form 2n - 1 also attracted attention because it is easy to show that if unless n is prime these number must be composite. These are often called Mersenne numbers Mn because Mersenne studied them.

Not all numbers of the form 2n - 1 with n prime are prime. For example 211 - 1 = 2047 = 23 89 is composite, though this was first noted as late as 1536.
For many years numbers of this form provided the largest known primes. The number M19 was proved to be prime by Cataldi in 1588 and this was the largest known prime for about 200 years until Euler proved that M31 is prime. This established the record for another century and when Lucas showed that M127 (which is a 39 digit number) is prime that took the record as far as the age of the electronic computer.
In 1952 the Mersenne numbers M521, M607, M1279, M2203 and M2281 were proved to be prime by Robinson using an early computer and the electronic age had begun.

By 2005 a total of 42 Mersenne primes have been found. The largest is M25964951 which has 7816230 decimal digits.

Euler's work had a great impact on number theory in general and on primes in particular.
He extended Fermat's Little Theorem and introduced the Euler φ-function. As mentioned above he factorised the 5th Fermat Number 232 + 1, he found 60 pairs of the amicable numbers referred to above, and he stated (but was unable to prove) what became known as the Law of Quadratic Reciprocity.
He was the first to realise that number theory could be studied using the tools of analysis and in so-doing founded the subject of Analytic Number Theory. He was able to show that not only is the so-called Harmonic series ∑ (1/n) divergent, but the series

1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ...

formed by summing the reciprocals of the prime numbers, is also divergent. The sum to n terms of the Harmonic series grows roughly like log(n), while the latter series diverges even more slowly like log[ log(n) ]. This means, for example, that summing the reciprocals of all the primes that have been listed, even by the most powerful computers, only gives a sum of about 4, but the series still diverges to ∞.

At first sight the primes seem to be distributed among the integers in rather a haphazard way. For example in the 100 numbers immediately before 10 000 000 there are 9 primes, while in the 100 numbers after there are only 2 primes. However, on a large scale, the way in which the primes are distributed is very regular. Legendre and Gauss both did extensive calculations of the density of primes. Gauss (who was a prodigious calculator) told a friend that whenever he had a spare 15 minutes he would spend it in counting the primes in a 'chiliad' (a range of 1000 numbers). By the end of his life it is estimated that he had counted all the primes up to about 3 million. Both Legendre and Gauss came to the conclusion that for large n the density of primes near n is about 1/log(n). Legendre gave an estimate for π(n) the number of primes ≤ n of

π(n) = n/(log(n) - 1.08366)

while Gauss's estimate is in terms of the logarithmic integral

π(n) = ∫ (1/log(t) dt where the range of integration is 2 to n.


You can see the Legendre estimate and the Gauss estimate and can compare them.

The statement that the density of primes is 1/log(n) is known as the Prime Number Theorem. Attempts to prove it continued throughout the 19th Century with notable progress being made by Chebyshev and Riemann who was able to relate the problem to something called the Riemann Hypothesis: a still unproved result about the zeros in the Complex plane of something called the Riemann zeta-function. The result was eventually proved (using powerful methods in Complex analysis) by Hadamard and de la Vallée Poussin in 1896.

There are still many open questions (some of them dating back hundreds of years) relating to prime numbers.
Some unsolved problems




The Twin Primes Conjecture that there are infinitely many pairs of primes only 2 apart.

Goldbach's Conjecture (made in a letter by C Goldbach to Euler in 1742) that every even integer greater than 2 can be written as the sum of two primes.

Are there infinitely many primes of the form n2 + 1 ?
(Dirichlet proved that every arithmetic progression : {a + bn | n N} with a, b coprime contains infinitely many primes.)

Is there always a prime between n2 and (n + 1)2 ?
(The fact that there is always a prime between n and 2n was called Bertrand's conjecture and was proved by Chebyshev.)

Are there infinitely many prime Fermat numbers? Indeed, are there any prime Fermat numbers after the fourth one?

Is there an arithmetic progression of consecutive primes for any given (finite) length? e.g. 251, 257, 263, 269 has length 4. The largest example known has length 10.

Are there infinitely many sets of 3 consecutive primes in arithmetic progression. (True if we omit the word consecutive.)

n2 - n + 41 is prime for 0 ≤ n ≤ 40. Are there infinitely many primes of this form? The same question applies to n2 - 79 n + 1601 which is prime for 0 ≤ n ≤ 79.

Are there infinitely many primes of the form n# + 1? (where n# is the product of all primes ≤ n.)

Are there infinitely many primes of the form n# - 1?

Are there infinitely many primes of the form n! + 1?

Are there infinitely many primes of the form n! - 1?

If p is a prime, is 2p - 1 always square free? i.e. not divisible by the square of a prime.

Does the Fibonacci sequence contain an infinite number of primes?

Here are the latest prime records that we know.

The largest known prime (found by GIMPS [Great Internet Mersenne Prime Search] in February 2005) is the 42nd Mersenne prime: M25964951 which has 7816230 decimal digits

The largest known twin primes are 242206083 238880 1. They have 11713 digits and were announced by Indlekofer and Ja'rai in November, 1995.

The largest known factorial prime (prime of the form n! 1) is 3610! - 1. It is a number of 11277 digits and was announced by Caldwell in 1993.

The largest known primorial prime (prime of the form n# 1 where n# is the product of all primes ≤ n) is 24029# + 1. It is a number of 10387 digits and was announced by Caldwell in 1993.

The development of group theory



The study of the development of a concept such as that of a group has certain difficulties. It would be wrong to say that since the non-zero rationals form a group under multiplication then the origin of the group concept must go back to the beginnings of mathematics. Rather we must take the view that group theory is the abstraction of ideas that were common to a number of major areas which were being studied essentially simultaneously.

The three main areas that were to give rise to group theory are:-




geometry at the beginning of the 19th Century,

number theory at the end of the 18th Century,

the theory of algebraic equations at the end of the 18th Century leading to the study of permutations.


(1) Geometry has been studied for a very long time so it is reasonable to ask what happened to geometry at the beginning of the 19th Century that was to contribute to the rise of the group concept. Geometry had began to lose its 'metric' character with projective and non-euclidean geometries being studied. Also the movement to study geometry in n dimensions led to an abstraction in geometry itself. The difference between metric and incidence geometry comes from the work of Monge, his student Carnot and perhaps most importantly the work of Poncelet. Non-euclidean geometry was studied by Lambert, Gauss, Lobachevsky and János Bolyai among others.
Möbius in 1827, although he was completely unaware of the group concept, began to classify geometries using the fact that a particular geometry studies properties invariant under a particular group. Steiner in 1832 studied notions of synthetic geometry which were to eventually become part of the study of transformation groups.

(2) In 1761 Euler studied modular arithmetic. In particular he examined the remainders of powers of a number modulo n. Although Euler's work is, of course, not stated in group theoretic terms he does provide an example of the decomposition of an abelian group into cosets of a subgroup. He also proves a special case of the order of a subgroup being a divisor of the order of the group.

Gauss in 1801 was to take Euler's work much further and gives a considerable amount of work on modular arithmetic which amounts to a fair amount of theory of abelian groups. He examines orders of elements and proves (although not in this notation) that there is a subgroup for every number dividing the order of a cyclic group. Gauss also examined other abelian groups. He looked at binary quadratic forms

ax2 + 2bxy + cy2 where a, b, c are integers.

Gauss examined the behaviour of forms under transformations and substitutions. He partitions forms into classes and then defines a composition on the classes. Gauss proves that the order of composition of three forms is immaterial so, in modern language, the associative law holds. In fact Gauss has a finite abelian group and later (in 1869) Schering, who edited Gauss's works, found a basis for this abelian group.

(3) Permutations were first studied by Lagrange in his 1770 paper on the theory of algebraic equations. Lagrange's main object was to find out why cubic and quartic equations could be solved algebraically. In studying the cubic, for example, Lagrange assumes the roots of a given cubic equation are x', x'' and x'''. Then, taking 1, w, w2 as the cube roots of unity, he examines the expression

R = x' + wx'' + w2x'''

and notes that it takes just two different values under the six permutations of the roots x', x'', x'''. Although the beginnings of permutation group theory can be seen in this work, Lagrange never composes his permutations so in some sense never discusses groups at all.

The first person to claim that equations of degree 5 could not be solved algebraically was Ruffini. In 1799 he published a work whose purpose was to demonstrate the insolubility of the general quintic equation. Ruffini's work is based on that of Lagrange but Ruffini introduces groups of permutations. These he calls permutazione and explicitly uses the closure property (the associative law always holds for permutations). Ruffini divides his permutazione into types, namely permutazione semplice which are cyclic groups in modern notation, and permutazione composta which are non-cyclic groups. The permutazione composta Ruffini divides into three types which in today's notation are intransitive groups, transitive imprimitive groups and transitive primitive groups.

Ruffini's proof of the insolubility of the quintic has some gaps and, disappointed with the lack of reaction to his paper Ruffini published further proofs. In a paper of 1802 he shows that the group of permutations associated with an irreducible equation is transitive taking his understanding well beyond that of Lagrange.

Cauchy played a major role in developing the theory of permutations. His first paper on the subject was in 1815 but at this stage Cauchy is motivated by permutations of roots of equations. However, in 1844, Cauchy published a major work which sets up the theory of permutations as a subject in its own right. He introduces the notation of powers, positive and negative, of permutations (with the power 0 giving the identity permutation), defines the order of a permutation, introduces cycle notation and used the term système des substitutions conjuguées for a group. Cauchy calls two permutations similar if they have the same cycle structure and proves that this is the same as the permutations being conjugate.

Abel, in 1824, gave the first accepted proof of the insolubility of the quintic, and he used the existing ideas on permutations of roots but little new in the development of group theory.

Galois in 1831 was the first to really understand that the algebraic solution of an equation was related to the structure of a group le groupe of permutations related to the equation. By 1832 Galois had discovered that special subgroups (now called normal subgroups) are fundamental. He calls the decomposition of a group into cosets of a subgroup a proper decomposition if the right and left coset decompositions coincide. Galois then shows that the non-abelian simple group of smallest order has order 60.

Galois' work was not known until Liouville published Galois' papers in 1846. Liouville saw clearly the connection between Cauchy's theory of permutations and Galois' work. However Liouville failed to grasp that the importance of Galois' work lay in the group concept.

Betti began in 1851 publishing work relating permutation theory and the theory of equations. In fact Betti was the first to prove that Galois' group associated with an equation was in fact a group of permutations in the modern sense. Serret published an important work discussing Galois' work, still without seeing the significance of the group concept.

Jordan, however, in papers of 1865, 1869 and 1870 shows that he realises the significance of groups of permutations. He defines isomorphism of permutation groups and proves the Jordan-Hölder theorem for permutation groups. Hölder was to prove it in the context of abstract groups in 1889.

Klein proposed the Erlangen Program in 1872 which was the group theoretic classification of geometry. Groups were certainly becoming centre stage in mathematics.

Perhaps the most remarkable development had come even before Betti's work. It was due to the English mathematician Cayley. As early as 1849 Cayley published a paper linking his ideas on permutations with Cauchy's. In 1854 Cayley wrote two papers which are remarkable for the insight they have of abstract groups. At that time the only known groups were groups of permutations and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication. He gives the 'Cayley tables' of some special permutation groups but, much more significantly for the introduction of the abstract group concept, he realised that matrices and quaternions were groups.

Cayley's papers of 1854 were so far ahead of their time that they had little impact. However when Cayley returned to the topic in 1878 with four papers on groups, one of them called The theory of groups, the time was right for the abstract group concept to move towards the centre of mathematical investigation. Cayley proved, among many other results, that every finite group can be represented as a group of permutations. Cayley's work prompted Hölder, in 1893, to investigate groups of order

p3, pq2, pqr and p4.

Frobenius and Netto (a student of Kronecker) carried the theory of groups forward. As far as the abstract concept is concerned, the next major contributor was von Dyck. von Dyck, who had obtained his doctorate under Klein's supervision then became Klein's assistant. Von Dyck, with fundamental papers in 1882 and 1883, constructed free groups and the definition of abstract groups in terms of generators and relations.

Group theory really came of age with the book by Burnside Theory of groups of finite order published in 1897. The two volume algebra book by Heinrich Weber (a student of Dedekind) Lehrbuch der Algebra published in 1895 and 1896 became a standard text. These books influenced the next generation of mathematicians to bring group theory into perhaps the most major theory of 20th Century mathematics.

The fundamental theorem of algebra



The Fundamental Theorem of Algebra (FTA) states

Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers.

In fact there are many equivalent formulations: for example that every real polynomial can be expressed as the product of real linear and real quadratic factors.

Early studies of equations by al-Khwarizmi (c 800) only allowed positive real roots and the FTA was not relevant. Cardan was the first to realise that one could work with quantities more general than the real numbers. This discovery was made in the course of studying a formula which gave the roots of a cubic equation. The formula when applied to the equation x3 = 15x + 4 gave an answer involving √-121 yet Cardan knew that the equation had x = 4 as a solution. He was able to manipulate with his 'complex numbers' to obtain the right answer yet he in no way understood his own mathematics.

Bombelli, in his Algebra, published in 1572, was to produce a proper set of rules for manipulating these 'complex numbers'. Descartes in 1637 says that one can 'imagine' for every equation of degree n, n roots but these imagined roots do not correspond to any real quantity.

Viète gave equations of degree n with n roots but the first claim that there are always n solutions was made by a Flemish mathematician Albert Girard in 1629 in L'invention en algèbre . However he does not assert that solutions are of the form a + bi, a, b real, so allows the possibility that solutions come from a larger number field than C. In fact this was to become the whole problem of the FTA for many years since mathematicians accepted Albert Girard's assertion as self-evident. They believed that a polynomial equation of degree n must have n roots, the problem was, they believed, to show that these roots were of the form a + bi, a, b real.

Now Harriot knew that a polynomial which vanishes at t has a root x - t but this did not become well known until stated by Descartes in 1637 in La géométrie, so Albert Girard did not have much of the background to understand the problem properly.

A 'proof' that the FTA was false was given by Leibniz in 1702 when he asserted that x4 + t4 could never be written as a product of two real quadratic factors. His mistake came in not realising that √i could be written in the form a + bi, a, b real.

Euler, in a 1742 correspondence with Nicolaus(II) Bernoulli and Goldbach, showed that the Leibniz counterexample was false.

D'Alembert in 1746 made the first serious attempt at a proof of the FTA. For a polynomial f he takes a real b, c so that f(b) = c. Now he shows that there are complex numbers z1 and w1 so that

|z1| < |c|, |w1| < |c|.

He then iterates the process to converge on a zero of f. His proof has several weaknesses. Firstly, he uses a lemma without proof which was proved in 1851 by Puiseau, but whose proof uses the FTA! Secondly, he did not have the necessary knowledge to use a compactness argument to give the final convergence. Despite this, the ideas in this proof are important.

Euler was soon able to prove that every real polynomial of degree n, n ≤ 6 had exactly n complex roots. In 1749 he attempted a proof of the general case, so he tried to proof the FTA for Real Polynomials:

Every polynomial of the nth degree with real coefficients has precisely n zeros in C.

His proof in Recherches sur les racines imaginaires des équations is based on decomposing a monic polynomial of degree 2n into the product of two monic polynomials of degree m = 2n-1. Then since an arbitrary polynomial can be converted to a monic polynomial by multiplying by axk for some k the theorem would follow by iterating the decomposition. Now Euler knew a fact which went back to Cardan in Ars Magna, or earlier, that a transformation could be applied to remove the second largest degree term of a polynomial. Hence he assumed that

x2m + Ax2m-2 + Bx2m-3 +. . . = (xm + txm-1 + gxm-2 + . . .)(xm - txm-1 + hxm-2 + . . .)

and then multiplied up and compared coefficients. This Euler claimed led to g, h, ... being rational functions of A, B, ..., t. All this was carried out in detail for n = 4, but the general case is only a sketch.

In 1772 Lagrange raised objections to Euler's proof. He objected that Euler's rational functions could lead to 0/0. Lagrange used his knowledge of permutations of roots to fill all the gaps in Euler's proof except that he was still assuming that the polynomial equation of degree n must have n roots of some kind so he could work with them and deduce properties, like eventually that they had the form a + bi, a, b real.

Laplace, in 1795, tried to prove the FTA using a completely different approach using the discriminant of a polynomial. His proof was very elegant and its only 'problem' was that again the existence of roots was assumed.

Gauss is usually credited with the first proof of the FTA. In his doctoral thesis of 1799 he presented his first proof and also his objections to the other proofs. He is undoubtedly the first t