1.3: Structural Arithmetic
Whenever the answer to a question turns out to be unexpectedly nice, one should ask oneself whether this is an accident, or whether there is some explanation which should perhaps have led one to expect such a result. For example:
- Exactly \(25\) of the integers up to \(100\) are prime numbers – and \(25\) is exactly one quarter of \(100\). This is certainly a beautifully memorable fact. But it is a numerical fluke, with no hidden mathematical explanation.
- \(11\) and \(101\) are prime numbers. Is this perhaps a way of generating lots of prime numbers:
\(11, 101, 1001, 10 001, 100 001, . . .?\)
It may at first be tempting to think so – until, that is, you remember what you found in Problem 6 (a)(iii).
Write out the first \(12\) or so powers of \(4\):
\(4, 16, 64, 256, 1024, 4096, 16384, 65536, ...\)
Now create two sequences:
the sequence of final digits: \(4, 6, 4, 6, 4, 6, . . .\)
the sequence of leading digits: \(4, 1, 6, 2, 1, 4, 1, 6, . . .\)
Both sequences seem to consist of a single “block”, which repeats over and over for ever.
(a) How long is the apparent repeating block for the first sequence? How long is the apparent repeating block for the second sequence?
(b) It may not be immediately clear whether either of these sequences really repeats forever. Nor may it be clear whether the two sequences are alike, or whether one is quite different from the other. Can you give a simple proof that one of these sequences recurs , that is, repeats forever?
(c) Can you explain why the other sequence seems to recur, and decide whether it really does recur forever?
The \(4\) by \(4\) “multiplication table” below is completely familiar.
\(\begin{array} &1&2&3&4 \\ 2&4&6&8 \\ 3&6&9&12 \\ 4&8&12&16\end{array}\)
What is the total of all the numbers in the \(4\) by \(4\) square? How should one write this answer in a way that makes the total obvious?