UIL NS and Mathematics Special Topics Session Special session for Not the Ordinary Types of Numbers (Part 2) UIL NS and Mathematics Special Topics Session Larry White Box 25 Millersview, Tx 76862 325-483-5446 [email protected] UIL State NS & Mathematics Contest Director UIL NS and Mathematics Special Topics Session Not the Ordinary Types of Numbers (Part 2) Special Numbers UIL NS and Mathematics Special Topics Session Perfect Numbers If the sum of the proper divisors of a number N is equal to N, then N is a perfect number. 6, 28, 496, 8,128, … These 4 perfect numbers were known by the Greeks long ago. UIL NS and Mathematics Special Topics Session Perfect Numbers 33,550,336 was discovered about 1460. 8,589,869,056 and 137,438,691,328 were publicized in 1988. UIL NS and Mathematics Special Topics Session Perfect Numbers Ø All perfect numbers found thus far are even numbers, but no proof that this is true exists, yet. Ø It appears that there is a finite number of perfect numbers, but that has not been proven, yet. Ø Even perfect numbers are also triangular numbers.This has been proven. UIL NS and Mathematics Special Topics Session Perfect Numbers Ø All perfect numbers except 6 are the sum of a series of consecutive odd cubes. Proven! 3 3 3 3 3 3 28 = 1 + 3 496 = 1 + 3 + 5 + 7 Ø There is the same number of even perfect numbers as the Mersenne primes …48 so far. Proven! Ø Every Mersenne prime can be used to generate a perfect number. Proven! UIL NS and Mathematics Special Topics Session Sublime Numbers If the total number or divisors is a perfect number and the sum of the divisors is a perfect number then it is a sublime number. ex. 12 … has 6 divisors … sum of the divisors is 28 UIL NS and Mathematics Special Topics Session Sublime Numbers Only 2 sublime numbers exist … 12 and 6,086,555,670,238,378,989,670,371,734,243, 169,622,657,830,773,351,885,970,528,324, 860,512,791,691,264 (A little bigger than 6 duodecilliard) UIL NS and Mathematics Special Topics Session Narcissistic Numbers If the sum of each of the digits of a number N raised to the power of the number of digits in N, then N is a narcissistic number. Let N = 153 (note there are 3 digits) 3 3 3 So, 153 = 1 + 5 + 3 = 1 + 125 + 27 UIL NS and Mathematics Special Topics Session Narcissistic Numbers Let N = 8,208 (note there are 4 digits) 4 4 4 4 So, 8,208 = 8 + 2 + 0 + 8 = 4,096 + 16 + 0 + 4,096 UIL NS and Mathematics Special Topics Session Narcissistic Numbers Ø There are only 88 narcissistic numbers. Proven! Ø The largest narcissistic number is: 115,132,219,018,763,992,565,095,597,973,971,522,401 (a little more than 115 unodecillion) UIL NS and Mathematics Special Topics Session Pretty Wild Narcissistic Numbers Numbers that can be generated using their own digits in order in any way that works. ex. 6,859 = (6 + 8 + 5)^(sqrt(9)) UIL NS and Mathematics Special Topics Session Pretty Wild Narcissistic Numbers 4 9 ex. 24,739 = 2 + 7! + 3 ex. 23,328 = 2 x 33! x 2 x 8 UIL NS and Mathematics Special Topics Session Vampire Numbers Numbers that can be generated using “fangs” … two numbers created from the digits of the original number. If the product of the fangs is the original number then it is considered to be a Vampire number. UIL NS and Mathematics Special Topics Session Vampire Numbers ex. 1,260 = 21 x 60 ex. 2,187 = 27 x 81 ex. 136,948 = 146 x 938 UIL NS and Mathematics Special Topics Session Harshad Numbers A number in base N that is divisible by the sum of its digits in base N. ex. 1,729 … 1+7+2+9 = 19 … 1729 ÷ 19 = 91 UIL NS and Mathematics Special Topics Session Harshad Numbers NOTE: 1,729 is a special Harshad Number The product of the sum of the digits and the digits of sum reversed is the number. 19 x 91 = 1729 UIL NS and Mathematics Special Topics Session Harshad Numbers Also note: 1,729 can be written as the sum of two cubes in two different ways. 3 3 1729 = 1 + 12 3 3 1729 = 9 + 10 UIL NS and Mathematics Special Topics Session Not the Ordinary Types of Numbers (Part 2) Math is not boring, especially “ ” recreational mathematics . Do some investigations and enjoy. HAVE A GREAT DAY! .