*Solutions to Exercises in the Introduction of How To Prove It by Daniel J Velleman.*

**Problem (2):** Make some conjectures about the values of for which is prime or the values of for which is prime. (You might start by making a table similar to Figure 1) .

**Solution:**

Okay, first lets make a table:

Prime? | Prime? | Prime? | |||
---|---|---|---|---|---|

2 | Yes | 8 | No | 5 | Yes |

3 | Yes | 26 | No | 19 | Yes |

4 | No | 80 | No | 65 | No |

5 | Yes | 242 | No | 211 | Yes |

6 | No | 728 | No | 665 | No |

7 | Yes | 2186 | No | 2059 | No |

8 | No | 6560 | No | 6305 | No |

Now let us make (up) some conjectures:

(1) For any value of , is not a prime number.

(2) For any value of , is a prime number if is a prime.

Note that conjecture 1 is true because will always be an even number. Conjecture 2 is not true since it breaks down when

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