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. Cheers Jim White http://www.sudokuxtra.com/forum/read.php?2,686,686#msg-686 Sun, 16 Aug 2026 16:39:04 +0100 Phorum 5.2.19 http://www.sudokuxtra.com/forum/read.php?2,686,694#msg-694 http://www.sudokuxtra.com/forum/read.php?2,686,694#msg-694 Christine Sudoku Xtra Discussion Forum Tue, 22 Feb 2011 21:00:25 +0000 http://www.sudokuxtra.com/forum/read.php?2,686,693#msg-693 http://www.sudokuxtra.com/forum/read.php?2,686,693#msg-693 gareth Sudoku Xtra Discussion Forum Tue, 22 Feb 2011 16:58:20 +0000 http://www.sudokuxtra.com/forum/read.php?2,686,692#msg-692 http://www.sudokuxtra.com/forum/read.php?2,686,692#msg-692
Gareth, just wondering hether the Lulu copy is due out soon? I do like to have a glossy magazine to solve in! :)]]> Christine Sudoku Xtra Discussion Forum Thu, 17 Feb 2011 07:12:53 +0000 http://www.sudokuxtra.com/forum/read.php?2,686,691#msg-691 http://www.sudokuxtra.com/forum/read.php?2,686,691#msg-691
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.]]> Mathimagics Sudoku Xtra Discussion Forum Fri, 11 Feb 2011 13:02:46 +0000 http://www.sudokuxtra.com/forum/read.php?2,686,690#msg-690 http://www.sudokuxtra.com/forum/read.php?2,686,690#msg-690 gareth Sudoku Xtra Discussion Forum Thu, 10 Feb 2011 10:37:39 +0000 http://www.sudokuxtra.com/forum/read.php?2,686,689#msg-689 http://www.sudokuxtra.com/forum/read.php?2,686,689#msg-689 Mathimagics Sudoku Xtra Discussion Forum Thu, 10 Feb 2011 04:30:47 +0000 http://www.sudokuxtra.com/forum/read.php?2,686,688#msg-688 http://www.sudokuxtra.com/forum/read.php?2,686,688#msg-688 gareth Sudoku Xtra Discussion Forum Wed, 09 Feb 2011 15:01:10 +0000 http://www.sudokuxtra.com/forum/read.php?2,686,687#msg-687 http://www.sudokuxtra.com/forum/read.php?2,686,687#msg-687
As you say, the numbers given are 'p' such that 2^p-1 × (2^p - 1) is a perfect number.]]> gareth Sudoku Xtra Discussion Forum Wed, 09 Feb 2011 14:48:25 +0000 http://www.sudokuxtra.com/forum/read.php?2,686,686#msg-686 http://www.sudokuxtra.com/forum/read.php?2,686,686#msg-686 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]]> Mathimagics Sudoku Xtra Discussion Forum Wed, 09 Feb 2011 03:08:42 +0000