Quiz (Prime Numbers)

(1) It has been conjectured that there are infinitely many primes of the form Exhibit five such examples.

(2) Give an example to show that the following conjecture is not true: Every positive integer can be written in the form , where is either a prime or 1, and

(3) Show that any prime of the form is also of the form

(4) Show that each integer of the form has a prime factor of this form.

(5) Show the only prime for which is  perfect square is

(6) Another unproven conjecture is that there are an infinitude of primes that are 1 less than a power of 2, such as Find four more of these primes.

(7) It has been conjectured that every even integer can be written as the difference of consecutive primes in finitely many ways. For example,



Express the integer 10 as the difference of two consecutive primes in 15 ways.

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