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
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!
28 = 13 + 33 496 = 13 + 33 + 53 + 73
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!