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Further investigation of the Mordell Equation

This post carries on from the previous post on the Mordell Equation – so make sure you read that one first – otherwise this may not make much sense.  The man pictured above (cite: Wikipedia) is Louis Mordell who studied the equations we are looking at today (and which now bear his name).

In the previous post I looked at solutions to the difference between a cube and a square giving an answer of 2.  This time I’ll try to generalise to the difference between a cube and a square giving an answer of k.  I’ll start with the same method as from the previous post:

In the last 2 lines we outline the 2 possibilities, either b = 1 or b = -1.  First let’s see what happens when b = 1:

This will only provide an integer solution for a if we have:

Which generates the following first few values for k when we run through m = 1, 2,3..:

k = 2, 11, 26, 47

We follow the same method for b = -1 and get the following:

Which generates the following first few values for k when we run through m = 1, 2,3…:

k = 4, 13, 28, 49

These are the values of k which we will be able to generate solutions to. Following the same method as in the previous post this generates the following solutions:

Let’s illustrate one of these results graphically.  If we take the solutions for k = 13, which are (17,70) and (17,-70), these points should be on the curve x cubed – y squared = 13.

This is indeed the case.  This graph also demonstrates how all solutions to these curves will have symmetrical solutions (e, f) and (e, -f).

We can run a quick computer program to show that this method does not find all the solutions for the given values of k, but it does ensure solutions will be found for the k values in these lists.

In the code solutions above, results are listed k, x, y, x cubed, y squared.  We can see for example that in the case of k = 11 our method did not find the solution x = 3 and y = 4 (though we found x = 15 and y = 58).  So, using this method we now have a way of finding some solutions for some values of k – we’ve not cracked the general case, but we have at least made a start!

Essential resources for IB students:

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Essential Resources for IB Teachers

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Essential Resources for both IB teachers and IB students

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