![]() ![]() We define this point as the point through which all lines with gradient -1 intersect. In this case we affix a special point at infinity to the Cartesian (x,y) plane. This will happen whenever the gradient of this line is -1, which will make it parallel to the graph’s asymptote y = -x. We now need to deal with the special case when a line joining 2 points on the curve does not intersect the curve again. We therefore define addition (our operation Θ) in this group as: In this case C’ is the point (46/3, -37/3) We then reflect this new point C in the line y = x, giving us C’. This will intersect the graph in a 3rd point, C (except in a special case to be looked at in a minute). We then take 2 coordinate points with rational coordinates (i.e coordinates that can be written as a fraction of integers). So, let’s see if we can establish a Abelian group based around the rational coordinates on our graph. (For example with the addition operation 1+2 = 2+1).Īs we have seen, the set of integers under the operation addition forms an abelian group. 0 is the identity element for addition)ĥ) Commutativity. (For example with the addition operation, 4+-4 = -4+4 = 0. For each a in A there exists a b in A such that a Θ b = b Θ a = e. (For example with the addition operation, (1+2) + 3 = 1 + (2+3) )Ĥ) Inverse. For all elements a,b,c in A, (a Θ b) Θ c = a Θ (b Θ c) (For example with the addition operation, the addition of 2 integers numbers is still an integer)ģ) Associativity. For all elements a,b in A, a Θ b = c, where c is also in A. a+0 = a for all a in the real numbers).Ģ) Closure. (for example 0 is the identity element for the addition operation for the set of integers numbers. If we start with a set A and and operation Θ.ġ) Identity. For an element e in A, we have a Θ e = a for all a in A. If we can establish the following rules hold then we can create an Abelian group. Establishing that a set is a group then allows certain properties to be inferred. Groups can be considered as sets which follow a set number of rules with regards to operations like multiplication, addition etc. We can see that both our integer solutions to this problem (1,12) and (9,10) lie on the curve: ![]() We will notice that the graph has a line of symmetry around y = x and also an asymptote at y = -x. The modern field of elliptical curve cryptography is closely related to the ideas below and provides a very secure method of encrypting data.įor some given integer value of A. The smallest number which can be formed through 3 distinct (positive) integer solutions to the equation is A = 87, 539, 319.Īlthough this began as a number theory problem it has close links with both graphs and group theory – and it is from these fields that mathematicians have gained a deeper understanding as to the nature of its solutions. In the case that A = 1729, we have 2 possible ways of finding distinct integer solutions: The general problem referenced above is finding integer solutions to the below equation for given values of A: Ramanujan was profoundly interested in number theory – the study of integers and patterns inherent within them. Ramanujan remarked in reply, ” No Hardy, it’s a very interesting number! It’s the smallest number expressible as the sum of 2 cubes in 2 different ways!” Visiting him in hospital, Hardy remarked that the taxi that had brought him to the hospital had a very “rather dull number” – number 1729. H Hardy led him to being invited to study in England, though whilst there he fell sick. His correspondence with the renowned mathematician G. The Indian mathematician Ramanujan (picture cite: Wikipedia) is renowned as one of great self-taught mathematical prodigies. Ramanujan’s Taxi Cabs and the Sum of 2 Cubes If you are a teacher then please also visit my new site: for over 2000+ pdf pages of resources for teaching IB maths! ![]()
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