1 / 13

Activity 1-6: Perfect Numbers and Mersenne Numbers

www.carom-maths.co.uk. Activity 1-6: Perfect Numbers and Mersenne Numbers. Task: pick a number n between 1 and 30, and write down all of its factors. What is s(n)?. Define s(n) to be the sum of all the factors of n except for n itself.

alexander-woodard
Download Presentation

Activity 1-6: Perfect Numbers and Mersenne Numbers

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. Task: pick a number n between 1 and 30, and write down all of its factors.What is s(n)? Define s(n) to be the sum of all the factors of n except for n itself. 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?

  3. 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?

  4. 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.

  5. 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.

  6. In 1876, a mathematician called Lucasproved 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.

  7. 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 - 2)2n-1 + 2n-1 = (2n-1 - 1)2n + 2n-1 = 2n-1(2n – 1).

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

  9. 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.

  10. 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

  11. 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.

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

More Related