A Perfect Number is an Integer Where the sum of all its
Exact Divisors add up to that Number.
The Number itself (although a Divisor) is not included in the sum, but 1 is.
Example :- The Number 6 is a Perfect Number its Divisors being, 1 + 2 + 3 = 6
Pythagoras noted that all Perfect Numbers exhibit several other
elegant properties.
Example:- Perfect Numbers are always the Sum of a Series of consecutive counting Numbers.
1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
He Also Linked "TWONESS" (Powers of 2) with Perfection as All Powers of 2 Fail by 1 to be
Perfect Numbers.
Euclid Discovers That Perfect Numbers are Always The Multiple of 2
Numbers, One of Which is A Power of 2 and The Other Being The Next Power of 2 minus 1.
Example.
21 x (22 - 1) = 2 x 3 = 6
22 x (23 - 1) = 4 x 7 = 28
24 x (25 - 1) = 16 x 31 = 496
26 x (27 - 1) = 64 x 127 = 8128
A Large Example Found By A Computer Using Euclid Rule was :-
2216,090 x (2216,091 - 1). This number if Computed has over 130,000 Digits.
An Abundant Number is an Integer where the sum of all its Exact
Divisors add up to any value greater than the Number.
Example :- The Number 24 is an Abundant Number its Divisors being,
1 + 2 + 3 + 4 + 6 + 8 + 12 = 36
A Defective Number is an Integer where the sum of all its Exact Divisors
add up to less than the Number.
Example :- The Number 44 is a Defective Number its Divisors being,
1 + 2 + 4 + 11 + 22 = 40
Note.. All Prime Numbers are Defective (1 is the only Divisor).
Two Numbers whose Divisors Add up to each other are known as
Amicable Numbers.
The Number 220 has Divisors which sum to 284 and the Number 284 has Divisors which Sum to 220.
Type in a Number, to prove Perfect, Abundant or Defective...
Remember :- Six is Perfect, Factorial, Triangular, Patrick McGoohan The Prisoner!
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