Killer Sudoku (Perfect?)

The numbers in that last puzzle (Issue 13, p16), i.e. the prime numbers 2,3,5,7,13,17,19,31,61 are not perfect, but are prime numbers for which there is a corresponding perfect number.

Perfect numbers are numbers like 6 and 28 which are the sum of all their proper divisors. So we have 6 = 1+2+3, and 28 = 1+2+4+7+14.

Now, each prime p in Gareth's list has the property that the Mersenne number 2^p - 1 is also prime. The resulting numbers are called Mersenne primes.

Thus 2⁷ - 1 = 127 which is prime, but 2¹¹ - 1 = 2047 is not prime (2047 = 23 x 89). So 11 is not in the list.

Each Mersenne prime 2^p - 1 then yields a perfect number when multiplied by 2^{p - 1}.

For example, with p = 3, we have 2³ - 1 = 7, 2^{p - 1} = 4, and 7 x 4 = 28, which is a perfect number.


Cheers
Jim White

Of course you're right. I half-edited the description of the numbers to fit on one line but didn't finish doing so - I will update the PDF.

As you say, the numbers given are 'p' such that 2^p-1 × (2^p - 1) is a perfect number.



Edited 4 time(s). Last edit at 02/09/2011 03:02PM by gareth.

I've updated the PDF, should anyone wish to fetch a version which correctly describes what these numbers are. (Note that this has no effect on how the puzzle actually solves, since it was just an item of trivia). Thanks very much to Jim for spotting this!

I excel at things trivial - one day I hope to master something useful!

Trivia, but not trivial. Isn't the search for perfect numbers one of the oldest problems in mathematics?

Yes, that formula for generating even perfect numbers was known to Euclid (c300BC).

Odd perfect numbers, on the other hand are strongly suspected of being non-existent, but nobody has yet been able to prove this. This is one of the great unsolved problems in number theory.

Prime numbers or perfect numbers I enjoyed solving this puzzle! Not too diffiicult either as all those 1s and 9s and 3s and 7s made the adding up quite easy!!

Gareth, just wondering hether the Lulu copy is due out soon? I do like to have a glossy magazine to solve in!

Sorry for the delay on the Lulu edition - I've dropped everything else to get three puzzle books finished by the end of the month. Sorry!!! (only one or two copies of each Lulu edition are bought each month)

Not a problem - I've got the pdf to keep me going! I think I must be your best customer!! I buy the pdf version each month, and also a copy from Lulu as well because I love to have a hard copy! I always have some unsolved puzzles at the end of each issue and keep going back to them to have another go! Who knows - they may be a collectors item some day, expecially if there are very few in existence!!!