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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: Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications. 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.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories. I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner). Please contact here for information on webinar training or for business ideas.

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### Getting a 7 in IB Maths Exploration Coursework Getting a 7 in IB Maths Exploration Coursework

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### IB Maths Exploration Guide IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

1. Investigation essentials,
2. Marking criteria guidance,
3. 70 hand picked interesting topics
4. Useful websites for use in the exploration,
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6. Avoiding common student mistakes
7. A selection of detailed exploration ideas
8. Advice on using Geogebra, Desmos and Tracker.

### IB HL Paper 3 Practice Questions (120 page pdf) IB HL Paper 3 Practice Questions

Seventeen full investigation questions – each one designed to last around 1 hour, and totaling around 40 pages and 600 marks worth of content.  There is also a fully typed up mark scheme.  Together this is around 120 pages of content.

### Modelling Guide

IB Exploration Modelling Guide

A 50 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Modelling Guide includes:

Linear regression and log linearization, quadratic regression and cubic regression, exponential and trigonometric regression, comprehensive technology guide for using Desmos and Tracker.

### Statistics Guide IB Exploration Statistics Guide

A 55 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Statistics Guide includes: Pearson’s Product investigation, Chi Squared investigation, Binomial distribution investigation, t-test investigation, sampling techniques, normal distribution investigation and how to effectively use Desmos to represent data. 