Posted by Mathimagics

Killer Sudoku (Perfect?) February 09, 2011 03:08AM |
Registered: 11 years ago Posts: 31 |

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 2^{7} - 1 = 127 which is prime, but 2^{11} - 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 2^{3} - 1 = 7, 2^{p - 1} = 4, and 7 x 4 = 28, which is a perfect number.

Cheers

Jim White

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

Thus 2

Each Mersenne prime 2

For example, with

Cheers

Jim White

Re: Killer Sudoku (Perfect?) February 09, 2011 02:48PM |
AdminRegistered: 12 years ago Posts: 377 |

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.

As you say, the numbers given are 'p' such that 2

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

Re: Killer Sudoku (Perfect?) February 09, 2011 03:01PM |
AdminRegistered: 12 years ago Posts: 377 |

Re: Killer Sudoku (Perfect?) February 10, 2011 04:30AM |
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Re: Killer Sudoku (Perfect?) February 10, 2011 10:37AM |
AdminRegistered: 12 years ago Posts: 377 |

Re: Killer Sudoku (Perfect?) February 11, 2011 01:02PM |
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Re: Killer Sudoku (Perfect?) February 17, 2011 07:12AM |
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Re: Killer Sudoku (Perfect?) February 22, 2011 04:58PM |
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Re: Killer Sudoku (Perfect?) February 22, 2011 09:00PM |
Registered: 12 years ago Posts: 195 |

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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