www.carom-maths.co.uk
Activity 1-6: Perfect Numbers
and Mersenne Numbers
Define s(n) to be
the sum of all the factors of n except for n itself.
Task: pick a number n between 1 and 30,
and write down all of its factors.
What is s(n)?
For example: 12 has the factors 1, 2, 3, 4, 6, and 12,
so s(12) is 16.
Does s(n) ever happen to equal n?
Are there any numbers between 1 and 30
for which this holds true?
If n = s(n)
then n is called a perfect number.
6 and 28 are perfect: 1 + 2 + 3 = 6, 1 + 2 + 4 = 7 + 14 = 28
Task: show that 496 is perfect.
Now a number than can be written as 2n - 1,
where n is a natural number,
is called a Mersenne number.
Task: find the first ten Mersenne numbers –
which of these are prime?
Conjecture: 2n-1 prime  n is prime
Certainly if n is composite, 2n - 1 is composite, since
2pq - 1 = (2p-1)(2p(q-1) + 2p(q-2) +... + 1).
So the arrow from left to right holds...
But...
211 - 1 = 2047 = 2389.
So the arrow from right to left does not hold.
Mersenne (who was a French mathematician
in the seventeenth century)
said that 2n - 1 would be prime for
n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257,
and for no other number less than 257.
He turned out to be not quite right
in saying this, but he was close!
The actual list is
2, 3, 5, 7, 13, 17, 19, 31, 69, 89, 107, 127.
Some of these numbers are huge, and until
computers it was very hard to check them.
In 1876, a mathematician called Lucas
proved that 2127 - 1 was prime,
and this remained the highest known prime for seventy years.
Nowadays the search for really large primes
still centres on Mersenne numbers.
And for every large Mersenne prime,
we have a large perfect number.
Task: Show that if m = 2n - 1 is prime,
then the mth triangle number
will be perfect.
The mth triangle number is
So if m = 2n - 1 is prime, then the mth triangle number’s
factors (not including itself) add to
1 + 2 + 4 +... + 2n-2 +2n-1
+ (1 + 2 + 4 + ... + 2n-2 )(2n-1)
= (1 + 2 + 4 + ... + 2n-2 )2n + 2n-1
= (2n-1 - 1)2n + 2n-1 = (2n - 2)2n-1 + 2n-1
= 2n-1(2n – 1).
It can be proved that all even perfect numbers are of this type.
What about the odd numbers?
It seems that s(n), where n is odd,
is always less than the number itself.
Is this true?
Task: try to find an odd number n
so that s(n) is greater than n.
What about 945?
945 = 33 x 5 x 7.
s(945) = 1+3+32+33+1x5+3x5+32x5+33x5
+1x7+3x7+32x7+33x7+1x5x7+3x5x7+32x5x7
= 975.
Searching with a spreadsheet is helpful...
Perfect Numbers and
Mersenne Numbers
spreadsheet
http://www.s253053503.websitehome.co.uk/
carom/carom-files/carom-1-6.xlsm
So for an odd number n
is it ever possible for n = s(n)?
Is there an odd perfect number?
People have looked extremely hard, but none have been found…
yet.
With thanks to:
William Dunham.
Carom is written by Jonny Griffiths, hello@jonny-griffiths.net