**Problem 1.1.1** Check that the quadratic function
is prime for all small values of
(say, for
up to
).

*Solution.* Using the matlab package in R, we check whether
the quantity
is prime for each positive natural number less than or
equal to
. We find that, for each
less than or equal to
, this quantity lacks a proper divisor other than
and is hence prime.

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Composite | ||

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**Problem 1.2.** Show nevertheless that
is not prime for certain values of
.

*Solution.* Let
. Then
meaning that
has two non-trivial proper divisor and is therefore
composite.

**Problem 1.3.** What is the smallest such value?

*Solution.* Using the matlab package in R and the table we
generated in Problem 1.1.1, we find that
is prime provided that
. But when
, we have
, which is composite. In particular, it has a proper
divisor of
as
. So
is the smallest such value of
for which
is not prime.