Apostol remarks that a similar argument shows that Ö{N2 +1} and Ö{N2 - 1} are irrational except in the obvious
cases.
In the above picture note that DEDC @ DABC, so
that
CD
CB
=
DE
AB
=
CE
AC
.
Irrationality of Ö{N2 + 1}
Suppose that Ö{N2 + 1} is a rational number.
If q2 (N2 + 1) = p2 for natural numbers q > 1, p > 1, we may construct DABC with integer sides so that
AC
= q
Ö
N2 + 1
,
AB
= q N,
CB
= q.
Choose the smallest such q . Then
CD
= q
æ è
Ö
N2 + 1
- N
ö ø
, aninteger.
Define
a
=
CD
CB
=
æ è
Ö
N2 + 1
- N
ö ø
.
Then
CD
= a CB = a q = q¢, aninteger,
DE
= a AB = a q N = q¢ N, aninteger,
CE
= BC - BE = BC - DE
= a AC = q¢
Ö
N2 + 1
, aninteger.
Irrationality of Ö{N2 - 1}
If q2 (N2 - 1) = p2 for natural numbers q > 1, p,
we may construct DABC with integer sides so that