A number that can be divided by one and itself
Think back to the last time you taught the objectives below:
- Recognise and use multiples, factors, primes (less than 100), common factors, highest common factors and lowest common multiples in simple cases; use simple tests of divisibility.
- Use multiples, factors, common factors, highest common actors, lowest common multiples and primes; find the prime factor decomposition of a number, eg 8 000 = 26 × 53.
- Use the prime factor decomposition of a number.
Learning Objectives as outlined within the Framework for Secondary Mathematics
How did it go? Did your students show the enthusiasm, intrigue and fascination demonstrated by countless number theorists over millennia or were the majority quite content at learning definitions, facts and a tidy mathematical procedure? It would be unreasonable to expect all students to see what has perplexed a large proportion of the great mathematicians over the years, but there must be something about prime numbers.
Despite their apparent simplicity and fundamental character, prime numbers remain the most mysterious objects studied by mathematicians. In a subject dedicated to finding patterns and order, the primes offer the ultimate challenge. Look through a list of prime numbers, and you’ll find that it’s impossible to predict when the next prime will appear. The list seems chaotic, random, and offers no clues as to how to determine the next number.
The Music of the Primes – Marcus du Sautoy
Since Euclid (c330-275 BC) discovered primes and proved that an infinite number exist, the riddle of predicting the pattern and finding a formula remains unsolved. The fact that this accessible series of numbers has no recognisable order seems to have driven the maths community to distraction. So is it possible to engage secondary school students in a similar way?
Take two different yet related questions:
A Calculate 29 x 31
B Which two prime numbers multiplied together give 437?
Question A is straightforward enough, especially with a calculator. Question B however requires a little more thought, mainly because trial and improvement is the only way to reach the answer of 19 x 23. Can you think of another calculation where the reverse operation is so much harder to calculate? Now imagine trying to solve a similar question where the prime numbers used are three, four, 10 or 1 000 digits long? The problem is deemed so difficult that e-businesses around the world have used the above principle to encode and secure their emails, finances and communications. The sender and receiver can scramble or decipher information using codes created by the multiplication of two large prime numbers. Without primes, our PIN numbers wouldn’t be as secure!
Primes have become part of people’s lives whether they know it or not. So is it not our job to tell them?
Some Prime Examples
|31 is prime
|331 is prime
|3331 is prime
|33331 is prime
|333331 is prime
|3333331 is prime
|33333331 is prime
|So surely 333333331 is prime?
|Well, apparently not. 333333331=17x19607843! And who said maths was logical?
Take the prime number 73939133.
Remove any of the digits from the end and the resulting number remains prime. No other known prime can boast this fact.
A good example of the use of prime numbers in nature is the evolutionary strategy used by cicadas.
‘Most cicadas go through a life cycle that lasts from two to five years. Some species have much longer life cycles, e.g., such as the North American genus, Magicicada, which has a number of distinct "broods" that go through either a 17-year or, in the American South, a 13-year life cycle. These long life cycles are an adaptation to predators such as the cicada killer wasp and praying mantis, as a predator could not regularly fall into synchrony with the cicadas. Both 13 and 17 are prime numbers, so while a cicada with a 15-year life cycle could be preyed upon by a predator with a three- or five-year life cycle, the 13- and 17-year cycles allow them to stop the predators falling into step.’
Cicada from Wikipedia
Prime Numbers Card Trick
‘Remove seven cards from a 52 card pack. (It’s probably best if they are consecutive and in the same suit, for example the Ace, 2, 3, 4, 5, 6 and 7 of Hearts.) Ask your assistant to check that these are ordinary cards. Now ask her to shuffle the cards, then take them back and shuffle them yourself. Secretly check the card at the bottom of the pile – let’s suppose that it is the Ace of Hearts.
Now tell the assistant that you have strong psychic powers, which enable you to prevent her from picking the Ace of Hearts. Give her the pile (face down) and ask her to think of any number between one and six. Suppose she picks four. Now tell her to count three cards from the top of the pack on to the bottom, one at a time, and then to turn over the top card. Predict that it won’t be the Ace of Hearts, and sure enough it isn’t. Ask her to place this card face up on the bottom of the pack and then repeat the exercise, counting three cards from the top on to the bottom one at a time, and turning over the fourth. She does this routine six times, and each time the card she turns over is not an Ace of Hearts. Only one card is left face down, and you tell her that, as usual, you managed to keep the selected card from appearing until the last moment. Turn is over to reveal the Ace of Hearts.
The only requirement for this card trick is that the number of cards in the pile is a prime number…....To anyone familiar with the principle of prime factors this result may be blindingly obvious, but it makes a surprisingly effective trick, even when performed on mathematicians!’
from Why do buses come in threes? – Rob Eastaway and Jeremy Wyndham