Last Updated：2026/01/01
Sentence
I
used
Euclid's
Lemma
in
a
slightly
sly
way
in
the
second
chapter,
where
I
ran
through
the
argument
that
√2
is
irrational.
I
said
there
that
if
2
is
a
factor
of
a²
then
a
itself
must
be
even.
This
follows
from
Euclid's
Lemma
upon
taking
p=2,
the
only
even
prime,
and
taking
b=a.
Indeed,
using
Euclid's
Lemma
it
is
not
hard
to
generalize
the
argument
showing
√2
to
be
irrational
to
prove
that
√p
is
irrational
for
any
prime
p.
Quizzes for review
I used Euclid's Lemma in a slightly sly way in the second chapter, where I ran through the argument that √2 is irrational. I said there that if 2 is a factor of a² then a itself must be even. This follows from Euclid's Lemma upon taking p=2, the only even prime, and taking b=a. Indeed, using Euclid's Lemma it is not hard to generalize the argument showing √2 to be irrational to prove that √p is irrational for any prime p.
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