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This page contains a list of special numbers, like prime, perfect, etc. It defines them and (when appropriate) has a program to generate them. Note: The programs are just to show how to generate the numbers, they will not get you in some hall of fame if you run them long enough because of accuracy problems;-)
Abundant numbers are numbers whose factors add up to more than the number. A good deal of even numbers are abundant, and very few odd numbers are(The first one is 945). Example: 12, factors are 1,2,3,4, and 6. 1+2+3+4+6=16 16>12.
Amicable numbers are a set of numbers whose factors(excluding themselves) add up to the other one. For example, 220 and 284 are the first set. 220's factors add up to 284 : 1+2+4+5+10+11+20+22+44+55+110=284 , and 284's factors add up to 220 : 1+2+4+71+142=220.The next ones are 1184 and 1210, 2620 and 2924, and so on. Click here to download the program in QBasic format, click here to download the program in executable format, click here to download the program in c format
Complex numbers are in the form a+bi, where i is the square root of negative one. They consist of a real part (a) and an imaginary part (b). For a long time, imaginary numbers where thought to be, well, imaginary. They were thought to have no purpose whatsoever in mathematics. Now, it turns out, there are many purposes, including problems that are unsolvable without them. A more aesthetic application is generating fractals, which are beautiful images such as can be seen by visiting The Fracint homepage(where you can download the program to make fractals) or by clicking this link.
Deficient numbers are numbers whose factors add up to less than the number. All primes are deficient, as are most low odd numbers (All odd numbers up until 945 are deficient). Example: 9, factors are 1 and 3 1 + 3=4 4<9.
Fermat primes are in the form of 22n+1, where n is a positive integer. Unfortunatly for Fermat (And math freaks everywhere who like to spout off large prime numbers), this only holds true for 1,2,3,and 4. 225=4294967297, and 4294967297 / 641=6700417. The four fermat primes are 5,17,257, and 65537.
Infinity, also denoted À0(Aleph Null), is an interesting 'number'. À0 is the 'number' of natural numbers (1,2,3,.....,À0). It's also the number of whole numbers, and the number of primes, etc.. But contrary to popular belief, the number of numbers between 0 and 1 is more than À0. It's À1. This is also the number of points on a line. À2 is the number of lines, curves, and other sets of points on a plane. There is an infinite number of these 'infinities', as proved by Georg Cantor a century ago. In fact, most of our common knowledge about infinity (very useful knowledge at that) came from his work.
Named after a french monk, these primes are in the form 2n-1 where n is a prime number. In other words, take two to the power of a prime number (n) and subtract one. Most of the time, 2n-1 isn't prime, but it gives a better guess at what might be a prime than just picking a random number. This equation generates enormously large prime numbers(In fact, the 6 largest know primes are mersenne primes, the largest having 909526 digits!). For example, 22-1 = 3 , 23-1 = 7 , 25-1 = 31 , 27-1 = 127, and so on. The first 5 Mersenne primes are 3,7,31,127, and 8191. Click here to download the program in QBasic format, click here to download the program in executable format, click here to download the program in c format
A palindrome is a number whose digits are mirror images of each other. 11, 454, and 332909233 are examples of palindromic numbers. People who have nothing better to do with their time can come up with some very strange things that these numbers do. 1*1=1, 11*11=121, 111*111=12321, 1111*1111=1234321, and so on. When you reach ten ones, it screws up with 1111111111*1111111111=1234567900987654321. In fact, if you multiply alot of small palindromes having small digits in them by 1 repeated a few times, you'll get another palindrome. For example: 3443*11=37873, 2211122*11111=24567776542, and 321232123*1111=356888888653. From Patrick De Geest's web page on palindromes: 123,456,789 * 989,010,989 = 122,100,120,987,654,321. Also, 1234567's prime factors are 127*9721, a palindrome of sorts. Put the two numbers together and subtract 1234567, and you get 45154 (1279721 - 1234567 = 45154)! Another palindrome.
A perfect number is a numbers whose factors besides itself add up to itself. For example, 6 is perfect: its factors are 1,2, and 3. 1+2+3=6. There are very few perfect numbers, this is because all perfect numbers are the sum of all the numbers from one to a mersenne prime. As just stated, 6 is a perfect number. The first mersenne prime is 3. 1+2+3=6. The next perfect number is 28. 28's factors are 1,2,4,7, and 14. 1+2+4+7+14=28. The next mersenne prime is 7. 1+2+3+4+5+6+7=28. Strange huh? The first 5 perfect numbers are 6,28,496,8128,and 33550336. A program to generate them is kinda worthless, because you can take the program for mersenne primes and use it to find perfect numbers. Don't want to type 1+2+3+4+5+6+7+8+9...+127 into a calculator? There's an easy way to compute that summation. The sum of the numbers from 1 to n is (n/2)*(n+1), simple huh? Even lazier? Click here to download the program in QBasic format, click here to download the program in executable format, click here to download the program in c format
Prime numbers are, as most people know, numbers that aren't divisible by any number but one and themselves. The first 5 prime numbers are 2,3,5,7 and 11. Click here to download the program in QBasic format, click here to download the program in executable format, click here to download the program in c format
