Perfect Numbers

Definitions

100

The prime numbers that result from Mp=2^n-1, n being prime, is called...

Mersenne Primes

200

A perfect number is a number that can be written as the ______ of two integers.

Sum

300

An ______ ______, n, is a number where the proper quotients have a sum greater than n.

Abundant number

400

_____ _______ is the concept of creating a sequence of double numbers until the sum is prime.

Double Proportions

History

100

Who wrote Proposition 36 of Book IX?

Euclid

200

_______ ______ was written around 300 B.C. and included the first recorded result about perfect numbers. 

Euclid's Elements

300

Perfect numbers (studied by Pythagoras and his followers) were thought to have what type of properties?

Mystical

400

This mathematician first named deficient and abundant numbers.

Nicomachus

Perfect Numbers

100

True or False: Euler successfully proved that all odd perfect numbers must follow the form (4n+1)4^k+1b².

False; there are no known odd perfect numbers to prove

200

True or False: Perfect numbers will always be prime numbers that satisfy 2ⁿ-1.

False

300

Fill in the blank: All known perfect numbers end in  ___ or ___.

6;28

400

 If k>1 and (2^k-1) is prime, then 2^k-1(2^k-1) is a _____ ______

Perfect number

Solve

100

Is 72 a perfect number?

No

200

Is 48 an abundant or deficient number?

Abundant

300

List the proper quotients of 96.

1,2,3,4,6,8,12,16,24,32,48

400

Show that 496 is a perfect number using double proportion *remember (sum x last)*

1+2+4+8+16=31

31(16)= 496