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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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.