Carr’s Synopsis and Sixth Term Entrance Papers

Students in search of extension work in maths, particularly as far as Sixth Term Entrance Papers (STEP) are concerned, should peruse Carr’s Synopsis of results in Elementary Pure Mathematics.

This book is famous for being the volume used by the autodidact Ramanujan to teach himself mathematics.  The book is merely a list of results and student how works through them, proving each one, will develop a strong ability in the subject.  This life line into higher mathematics could not be produced in the same form in today’s educational and publishing context but, perhaps, Carr deserves a place as one of the subject’s great educators.

Higher mathematics is about generalisation and abstraction.  Because of this, school mathematics students can find university text books unreadable.  In such a context how can harder, higher level questions be set using the frame work of sixth form mathematics? The answer to this question is to wind the clock back to the 1880s and see how the subject was constructed then.  Carr does this for us; many useful stages in school mathematics are listed here with extensions and generalisation written in generally sixth form recognisable form.

Take question 1 from this year’s (2016) STEP I exam.  Pupils will only have to be familiar with the first six results to tackle this quickly.

carr1to6

The question play is all about the divisibility of,

x^{2n+1}+1

which result 6, above, tells us is

(x+1)(x^{2n}-x^{2n-1}+\dots+1).

Question game play leads us to factorise large numbers (without use of calculator):

\dfrac{300^{3}+1}{301}=89911\times 90091

and

\dfrac{7^{49}+1}{7^{7}+1}=[(7^{7}+1)^{3}-7^{4}(7^{14}+7^{7}+1)][(7^{7}+1)^{3}+7^{4}(7^{14}+7^{7}+1)]

in a difference of two squares (result 1) construction.

Full solution is attached.

step-i-2016-q1

I think that mathematically interested sixth formers, wanting to study the subject at University, should have some background knowledge.  Many would have heard about the Riemann Hypothesis and a link to prime numbers; this subject is beyond the scope of this blog but school pupils, with knowledge of prime numbers and geometric progressions, can see how the Zeta Function relates to prime numbers in the above bit of mathematics.  I like it anyway.