Perfect numbers

Factor chains Here’s another one: 9 10 Look at the chain of numbers. How do you think the chain is made? 4 8 3 7 1 1

Factor chains 10 1, 2, 5, 10 1 + 2 + 5 = 8 1, 2, 4, 8 1 + 2 + 4 = 7 1, 7 1 = 1 1 End! Factor chains are made by listing all of the factors of a number, and summing them (excluding the original number). Can you see how this makes the chain opposite? 8 7 1

Factor chains Complete the factor chains for the numbers below: 15 16 – 9 – 4 – 3 – 1 – 15 – 9 – 4 – 3 – 1 13 – 1 22 – 14 – 10 – 4 – 3 – 1

Factor chains- what did you notice? Questions to consider: Do all of the chains end in 1? Is the penultimate number always prime? Are the numbers in the chain always decreasing? What is the shortest chain possible? What is the longest chain you can find? Pick your own staring numbers and investigate.

An unusual number chain! What happens when you start with 6? Or 25? 6 25 It turns out 6 is a pretty special number- it is a PERFECT NUMBER. There is one other number between 1-30 like this, can you find it?

Perfect numbers A perfect number is a positive integer that is equal to the sum of its factors, excluding the number itself. You can also think of perfect numbers as half the sum of all of its factors (including itself).

Perfect numbers Perfect numbers have fascinated mathematicians for a very long time. Even Greek mathematician Euclid, born in 325BC, wrote about them. It is not known whether there are any odd perfect numbers, or whether infinitely many perfect numbers exist. Finding them can be tricky. We know 6 and 28, but what is the next perfect number?

The hunt for perfect numbers It was thought that the sequence below might help find perfect numbers, can you see how? 1 3 7 … Can this help? What could the next term in the sequence be? What possible perfect number does that give? Adding 6 gives 13, 6x13 gives 78 which isn’t a perfect number Adding 8 gives 15, 8x15 gives 120 which isn’t a perfect number D’oh! +2 +4 3x2=6 7x4=28

The third perfect number The third perfect number is actually 496. Can we convince ourselves 496 is perfect?

The hunt for perfect numbers It was thought that the sequence below might help find perfect numbers. It didn’t seem to work- but can it? 1 3 7 15 31 63 … +2 +4 +16 +32 +8 If the term is prime, is seems a perfect number can be generated. The next prime in the sequence is 127, giving 8128- which is perfect! 3x2=6 7x4=28 15x8=120 31x16=496 The terms 3, 7 and 31 produced a perfect number, but 15 didn’t. What do you think the next perfect number is?

Euclid-Euler Perfect numbers Around 2000 years after Euclid wrote about perfect numbers, another famous mathematician, Leonhard Euler, was thinking about them. Take any prime number, p, Euler proved that whenever 2p − 1 is prime, then 2p−1(2p − 1) is a perfect number- just like in our sequence!

Euclid-Euler Perfect numbers Using this formula 2p−1(2p − 1), the first four perfect numbers can be generated as follows: p = 2: 21(22 − 1) = 6 p = 3: 22(23 − 1) = 28 p = 5: 24(25 − 1) = 496 p = 7: 26(27 − 1) = 8128 The hunt for perfect numbers is seemingly over- BUT mathematicians still don’t know if an ODD perfect number exists, or how many perfect numbers there are!

Perfect numbers 10/11/2017