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Activity 1-6: Perfect Numbers and Mersenne Numbers www.carom-maths.co.uk

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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? Define s(n) to be the sum of all the factors of n except for n itself.

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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 2 n - 1, where n is a natural number, is called a Mersenne number. Task: find the first ten Mersenne numbers – which of these are prime?

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Conjecture: 2 n -1 prime n is prime Certainly if n is composite, 2 n - 1 is composite, since 2 pq - 1 = (2 p -1)(2 p(q-1) + 2 p(q-2) +... + 1). But... 2 11 - 1 = 2047 = 23 89. So the arrow from left to right holds... So the arrow from right to left does not hold.

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Mersenne (who was a French mathematician in the seventeenth century) said that 2 n - 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.

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In 1876, a mathematician called Lucas proved that 2 127 - 1 was prime, and this remained the highest known prime for seventy years. Task: Show that if m = 2 n - 1 is prime, then the m th triangle number will be perfect. Nowadays the search for really large primes still centres on Mersenne numbers. And for every large Mersenne prime, we have a large perfect number.

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The m th triangle number is 1 + 2 + 4 +... + 2 n-2 +2 n-1 + (1 + 2 + 4 +... + 2 n-2 )(2 n -1) = (1 + 2 + 4 +... + 2 n-2 )2 n + 2 n-1 = (2 n-1 - 1)2 n + 2 n-1 = 2 n-1 (2 n – 1). So if m = 2 n - 1 is prime, then the m th triangle number’s factors (not including itself) add to = (2 n - 2)2 n-1 + 2 n-1

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It can be proved that all even perfect numbers are of this type. What about the odd numbers?

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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.

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What about 945?945 = 3 3 x 5 x 7. s(945) = 1+3+3 2 +3 3 +1x5+3x5+3 2 x5+3 3 x5 +1x7+3x7+3 2 x7+3 3 x7+1x5x7+3x5x7+3 2 x5x7 = 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

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People have looked extremely hard, but none have been found… So for an odd number n is it ever possible for n = s(n)? Is there an odd perfect number? yet.

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With thanks to: William Dunham. Carom is written by Jonny Griffiths, hello@jonny-griffiths.nethello@jonny-griffiths.net

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