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Killer Sudoku (Perfect?)

Posted by Mathimagics 
Killer Sudoku (Perfect?)
February 09, 2011 03:08AM
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 2p - 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 2p - 1 then yields a perfect number when multiplied by 2p - 1.

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

Jim White
Re: Killer Sudoku (Perfect?)
February 09, 2011 02:48PM
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 2p-1 × (2p - 1) is a perfect number.

Edited 4 time(s). Last edit at 02/09/2011 03:02PM by gareth.
Re: Killer Sudoku (Perfect?)
February 09, 2011 03:01PM
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!
Re: Killer Sudoku (Perfect?)
February 10, 2011 04:30AM
I excel at things trivial - one day I hope to master something useful! smoking smiley
Re: Killer Sudoku (Perfect?)
February 10, 2011 10:37AM
Trivia, but not trivial. winking smiley Isn't the search for perfect numbers one of the oldest problems in mathematics? smiling smiley
Re: Killer Sudoku (Perfect?)
February 11, 2011 01:02PM
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.
Re: Killer Sudoku (Perfect?)
February 17, 2011 07:12AM
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! smiling smiley
Re: Killer Sudoku (Perfect?)
February 22, 2011 04:58PM
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)
Re: Killer Sudoku (Perfect?)
February 22, 2011 09:00PM
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!!!
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