Page 141 - 'Blast_Into_Math

Blast into Math! Mathematical perspectives: all aour mase are melong to us

p
• Hint for # 4: You have seen that every even perfect number is of the form 2 p−1 (2 − 1) for
a prime number p such that 2 − 1 is a Mersenne prime. What does this look like when
p
you write it in base 2? You may find #2 and # 3 helpful.
• Hint for #8: If the base b is even, then there is some x ∈ N such that

b =2x.

x
So, what is ? If the base b is odd, then,
b
b − 1 1 (b − 1)/2
> > .
b 2 b

What do you do next?

• Hint for # 9: If you need some inspiration, listen to “Walk like an Egyptian” by the Bangles.
• Hint for # 10: What is a set of rational numbers whose least upper bound should be
√
something not in Q? Think about the definition of square root: x is the number whose
square is equal to x . If two rational numbers x and y satisfy

0 <x<y,

then

2
2
0 <x <y .

Use this and the something you proved is not in Q to construct a set of numbers which are
in Q and are less than that something.

• Hint for # 11: Once you have made the induction assumption, that the theorem is true for
some n ≥ 3, then the exponent rules tell you that

n
(a + b) n+1 =(a + b)(a + b) ,

which by the induction assumption means

n n! n n!
k n−k
n
j n−j
(a + b) n+1 =(a + b)(a + b) = a a b + b a b .
k!(n − k)! j!(n − j)!
k=0 j=0

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