### Project 5.1

In the text we tried some formulas that might have generated only primes, but, alas, they failed. Below are some other formulas. Show that these, too, do not generate only primes.

**a.** *n ^{2} + n* + 41

**b.**

*n*79

^{2}−*n*+ 1,601

**c.**2

*n*+ 29

^{2}**d.**9

*n*498

^{2}−*n*+ 6,683

**e.**

*n*+ 1,

^{2}*n*an even integer

### Project 5.2

For what values of *n* is 11*14^{n} + 1 a prime?

*Hint*: Consider *n* even, and then consider *n* odd.

### Project 5.3

A formula that generates all prime numbers is given by David Dunlop and Thomas Sigmundin their book *Problem Solving with the Programmable Calculator* (Englewood Cliffs, N.J.: Prentice-Hall, 1983). The authors claim that the formula square root of (1 + 24*n*) produces every prime number except 2 and 3, but give no proof or reference to a proof. Create a table, and give an argument to support or find a counterexample to disprove their claim.

### Project 5.4

A large prime, 2^{30,402,457} , is a number that has 9,152,052 digits. A number this large is hard to comprehend. Write a paper making the size of this number meaningful to a non-mathematical reader.

### Project 5.5

Investigate some of the properties of primes not discussed in the text. Why are primes important to mathematicians? Why are primes important in mathematics? What are some of the important theorems concerning primes?

**Reference:****Martin Gardner**, “The Remarkable Lore of Prime Numbers,” *Scientific American,* March 1964.