Year: 2014
Paper: 1
Question Number: 1
Course: LFM Pure
Section: Proof
More than 1800 candidates sat this paper, which represents another increase in uptake for this STEP paper. The impression given, however, is that many of these extra candidates are just not sufficiently well prepared for questions which are not structured in the same way as are the A-level questions that they are, perhaps, more accustomed to seeing. Although STEP questions try to give all able candidates "a bit of an intro." into each question, they are not intended to be easy, and (at some point) imagination and real flair (as well as determination) are required if one is to score well on them. In general, it is simply not possible to get very far into a question without making some attempt to think about what is actually going on in the situation presented therein; and those students who expect to be told exactly what to do at each stage of a process are in for a shock. Too many candidates only attempt the first parts of many questions, restricting themselves to 3-6 marks on each, rather than trying to get to grips with substantial portions of work – the readiness to give up and try to find something else that is "easy pickings" seldom allows such candidates to acquire more than 40 marks (as was the case with almost half of this year's candidature, in fact). Poor preparation was strongly in evidence – curve-sketching skills were weak, inequalities very poorly handled, algebraic capabilities (especially in non-standard settings) were often pretty poor, and the ability to get to grips with extended bits of working lacking in the extreme; also, an unwillingness to be imaginative and creative, allied with a lack of thoroughness and attention to detail, made this a disappointing (and, possibly, very uncomfortable) experience for many of those students who took the paper. On the other side of the coin, there was a very pleasing number of candidates who produced exceptional pieces of work on 5 or 6 questions, and thus scored very highly indeed on the paper overall. Around 100 of them scored 90+ marks of the 120 available, and they should be very proud of their performance – it is a significant and noteworthy achievement.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\textit{All numbers referred to in this question are non-negative integers.}
\begin{questionparts}
\item Express each of the numbers 3, 5, 8, 12 and 16 as the difference of two non-zero squares.
\item Prove that any odd number can be written as the difference of two squares.
\item Prove that all numbers of the form $4k$, where $k$ is a non-negative integer, can be written as the difference of two squares.
\item Prove that no number of the form $4k+2$, where $k$ is a non-negative integer, can be written as the difference of two squares.
\item Prove that any number of the form $pq$, where $p$ and $q$ are prime numbers greater than 2, can be written as the difference of two squares in exactly two distinct ways. Does this result hold if $p$ is a prime greater than 2 and $q=2$?
\item Determine the number of distinct ways in which 675 can be written as the difference of two squares.
\end{questionparts}\begin{questionparts}
\item $\,$ \begin{align*}
&& 3 &= 2^2 - 1^2 \\
&& 5 &= 3^2 - 2^2 \\
&& 8 &= 3^2 - 1^2 \\
&& 16 &= 5^2 - 3^2
\end{align*}
\item Suppose $n = 2k+1$, then $n = (k+1)^2 - k^2$
\item Suppose $n = 4k$ then $n = (2k+1)^2 - (2k-1)^2$
\item All squares leave a remainder of $0$ or $1$ on division by $4$. Therefore the difference can leave a remainder of $0$, $1$, $-1 \equiv 3$, none of which are $2$.
\item Suppose $n = pq = a^2 - b^2$ with $a > b$ ie $(a-b)(a+b) = pq$. Since $p$ is prime, $p \mid (a-b)$ or $p \mid (a+b)$. Similarly for $q$. Suppose also (wlog) that $p > q$
Since the factors of $pq$ are $1, p, q, pq$ then $a-b = 1, p$ (which are two possibilities) and $a+b = pq, q$, ie $a = \frac{1+pq}{2}, \frac{p+q}{2}$ and $b = \frac{pq-1}{2}, \frac{p-q}{2}$
\begin{align*}
&& pq &= \left ( \frac{1+pq}{2} \right)^2- \left ( \frac{1-pq}{2} \right)^2 \\
&&&= \left ( \frac{p+q}{2} \right)^2- \left ( \frac{p-q}{2} \right)^2 \\
\end{align*}
Where everything is an integer since $p$ and $q$ are odd.
If we have $p > 2$ and $q = 2$ then $p$ is odd and the number has the form $4k+2$ which cannot be expressed as the difference of two squares.
\item $675 = 3^3 \cdot 5^2$, each factor pair of $675$ will lead to a different solution of $675 = a^2-b^2$, since we will have an equation $a-b = X, a+b = Y$ where $X, Y$ are both odd. Therefore there are as many solution as (half) the number of factors, ie $4 \times 3 = 12$
\end{questionparts}
Traditionally, question 1 is intended to be the most generous and/or helpful question on the paper, in order to permit as many candidates as possible to get started in a reasonably friendly situation, and thereby pick up at least 10 marks on the paper; giving them a positive start to the examination. This year, however, despite the high rate of popularity (over 80% of the candidature attempted Q1), there were several surprises in store for the examiners. Firstly, it was not nearly so popular as it appears from the proportion of attempts, as it turned out that many of these attempts were either weak or inconsequential, petering out the moment the work became algebraic rather than numerical. The other surprise was how poorly the very simple ideas were handled. Many candidates clearly did not know what constituted a proof in these settings, when little more than (say) a statement such as 2k + 1 = (k + 1)² − k² in (ii) was perfectly sufficient. Quite a few went on to attempt what was clearly intended to be an inductive proof, having already written the correct (and wholly adequate) result, sometimes in each of parts (ii), (iii) and (iv). Furthermore these sorts of mistakes were often preceded by incorrect numerical work in part (i), including offerings that ignored the initial prompting regarding the use of non-negative integers, such as "12 = 7² − 37". In other parts of the question, candidates would resort to providing counterexamples to results that had not been suggested; such as, in part (iv) producing a counterexample (such as "6") to refute the notion that "every number of the form 4k + 2 can be written as the difference of two squares" when the question actually required them to show that no number of this form has the proposed property, so offering one example represented a considerable misunderstanding of mathematical ideas and terminology. Part (iv) also suffered from the common misconception that factorising 4k + 2 as 2(2k + 1) immediately meant that 2k + 1 had to be prime. Candidates who had seen and used modular arithmetic had a bit of an advantage in (iv) but, in fact, there was very little evidence of such. Partly balancing the widespread lack of (pretty basic) number theoretic appreciation were the few candidates who found this all very straightforward, as had been intended to be the case. Overall, however, this question provided a very disappointing range of responses, and the mean score of under 5/20 underlines this fact.