1. Let p be a prime number. Let n be a natural number. Consider is a circular configuration of p equally spaced beads. Each of the p beads has n possible colors (it is allowed that multiple beads. Two necklaces are the same if they can be rotated. How many distinct necklaces can be made with p beads? Deduce Fermat's little theorem. 2. Still let p be a prime number and n be a natural number. A Bracelet is like a necklace except two bracelets are also considered the same if you can reflect one and get the other. How many possible bracelets are there. 3. Color the 6 edges of a regular tetrahedron with n colors (again with repetition allowed). Consider 2 colorings equivalent if you can rotate one tetrahedron onto the other. How many possible colorings are there. (A good first step would be to figure out all the possible rotations which send a regular tetrahedron to a regular tetrahedron, you know how many there are from a previous problem)
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Theorem 2.3 (Infinitude of primes): There are infinitely many prime numbers. PROOF: Suppose you have a finite list of prime numbers. Multiply all the prime numbers in your list together and call the result N. So, N is a positive integer. As N + 1 is again a positive integer bigger than 1, N + 1 has a prime factor p. Remember, you began with a list of prime numbers! If p were in your list, then p divides N (since N was the product of primes in your list) and p divides N + 1 (we chose p to be a factor of N + 1). By the two-out-of-three principle, p divides (N + 1) - N, so p divides 1. But no prime number is a factor of 1! Hence, the hypothesis ("if p were in the list") cannot be true. The prime number p is a "new prime," not in the list. Since no finite list of prime numbers contains all the prime numbers, there are infinitely many prime numbers.
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