d.

How could you use the above theorem to correct errors in a transmitted ISBN?
Explain.

e.

What mathematical ideas do you suppose were used to develop the above error-
correcting algorithm? (To get you started in your thinking, remember that Gauss
introduced the idea of congruence modulo n.) Is there an interplay between pure
and applied mathematics? Explain.

2.

The Euler

ϕ

-function

Recall that two integers are relatively primeif the only divisor they have in common is
±1.

We have already discussed Euclid’s proof that there are infinitely many prime numbers.
But have you ever wondered about the distribution of the primes? For example, how
many prime numbers are there between 1 and 100? or between 1 and 1000?

You probably agree that these seem like idle, though possibly interesting, questions.
Here is another one in the same spirit: Given a natural number n, how many integers
from

to nare relatively prime to n?

Leonhard Euler became interested in this question, and to study it, he defined what we
now call the Euler ϕ -function (pronounced “fee-function”). If n is a natural number,
then ϕ (n) equals the number of integers from

to nthat are relatively prime to n.

Answer the questions below.

a.

Verify that

ϕ

(1)

=1,ϕ(2)

=1,ϕ(3)

=2,ϕ( 4)

=2,ϕ(5)

=4.

b.

Find

ϕ6),ϕ( 7),

(

ϕ8),ϕ(9),

(

ϕ

(10),ϕ

(11),ϕ

(12),ϕ

(13),ϕ

(14),ϕ

(15).

c.

If nis a natural number and ais relatively prime to n, then Euler generalized a
theorem of Fermat and showed (i.e. proved) that aϕ(n)≡1(modn). Verify this
theorem for n=5and n=14.

d.

Would you guess that Euler’s theorem (stated in part c.), a theorem of pure
mathematics, had any application to real-world problems? Discuss.

3.

A Mathematician’s Apology

In 1940, the eminent mathematician G.H. Hardy published A Mathematician’s

Apology. The book was published subsequently in 1941, 1948, and 1967. We have

prepared a handout from this book for today’s discussion. The handout is a Xerox of
pages 59 and 60, from section 21 of the 1941 edition.

Read this selection and discuss it in light of the above mathematical examples and in
light of what you know about pure and applied mathematics. The following quote is
particularly pertinent: “The ‘real’ mathematics of the ‘real’ mathematicians, the
mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly
‘useless’...”