Perfect NumbersDate: 12/07/96 at 23:51:42 From: Ken Subject: Perfect Number What are the first 10 perfect numbers? Is there a formula for getting a perfect number? Date: 01/26/97 at 16:26:16 From: Doctor Reno Subject: Re: Perfect Number Hi, Ken! Perfect numbers are something that I recently became interested in. They have a fascinating history in mathematics. My response will be somewhat detailed due to the richness of the material. With this information, you will be able to figure out how to find all of the perfect numbers that you want to find! By definition, a perfect number is a number which equals the sum of its factors. In other words, 6 is perfect because the factors of 6 are 1, 2 and 3 and 1 + 2 + 3 = 6. 28 is the next perfect number because its factors are 1 + 2 + 4 + 7 + 14, whose sum is 28. Over two thousand years ago, Euclid wrote about constructing perfect numbers in this way: "If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect." This sounds complicated, but if we take it step by step, we can understand what Euclid was talking about! I will take phrases from this long sentence and we will explore what is happening. (In case the ancient lingo is a little confusing, a "unit" is 1, and "double proportion" means doubling, or multiplying by 2.) "an unit be set out continuously in double proportion": 1,2,4,8,16,32.... (1 x 2 = 2, 2 x 2 = 4, 4 x 2 = 8, etc) ...and Euclid says we can do this as many times as we please! "if the sum": easy! we can do this one! 1 + 2 + 4 + 8 = 16 "until the sum of all becomes prime": simply means that we add until we get to a prime number! 1 + 2 = 3 That's easy, isn't it! "the sum multiplied into the last make some number" Easier than it sounds: simply multiply the sum to the last number: 3 x 2 = 6 "the product will be perfect" There you have it! 6 is the first perfect number! Using this method, we can easily find the next perfect number: 1 + 2 + 4 = 7 7 x 4 = 28: the second perfect number! And the next...... 1 + 2 + 4 + 8 = 15....oops! 15 is NOT prime, so we have to go on. 1 + 2 + 4 + 8 + 16 = 31 OK! 31 is prime, so we can use Euclid's method again. 31 x 16 = 496 The third perfect number. And the next? 1 + 2 + 4 + 8 + 16 + 32 = 63 63 is NOT PRIME 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 127 is prime (it would help to have a list of primes, wouldn't it?!) 127 x 64 = 8,128 (a calculator will help, too!) The fifth perfect number (I've done a little of the work for you): 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048 + 4096 = 8191 8191 x 4096 = 33,550,336 We now have the first five perfect numbers: 6 28 496 8128 33,550,336 As you can see, it will be difficult to find the first ten perfect numbers....they will get very, very large! If you need a table of prime numbers, I have found one at: http://www.utm.edu/research/primes/lists/small/10000.txt Do you see some patterns in these numbers? Many mathematicians throughout history have been fascinated by the patterns that you also see! They have played with perfect numbers and come up with a lot of different ideas about them. The formula for finding perfect numbers is: 2^n-1(2^n - 1), where (2^n - 1) is prime. It is not known at this time whether there are an infinite number of perfect numbers or not. Here is a web site that gives some history and references for perfect numbers: MathLand: http://www.sciencenews.org/sn_arc97/1_25_97/mathland.htm I hope you have as much fun finding your perfect numbers as I had helping you with your questions! -Doctor Reno, The Math Forum |
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