3. (15 points)¹ A sexy prime pair is an ordered pair (p, q) of primes such that q- p = 6. For example, the first sexy pr

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3 15 Points A Sexy Prime Pair Is An Ordered Pair P Q Of Primes Such That Q P 6 For Example The First Sexy Pr 1 (156.63 KiB) Viewed 48 times

3. (15 points)¹ A sexy prime pair is an ordered pair (p, q) of primes such that q- p = 6. For example, the first sexy prime pair is (5,11) (we order sexy prime pairs by their first entry). (a) Determine the first five sexy prime pairs (starting with the one we gave). Show all of your work (i.e., the pairs you check, which ones work, and which ones do not). (b) A sexy prime quadruple is an ordered quadruple (p, q, r, s) of primes such that (p, q), (q, r), and (r, s) are all sexy prime pairs. For example, the first sexy prime quadruple is (5, 11, 17, 23) (we order sexy prime quadruples by their first entry). Prove that, if p> 5 and (p, q, r, s) is a sexy prime quadruple, then p= 1 (mod 10). (Hint: consider all possibilities!) (c) Determine the first four sexy prime quadruples (starting with the one we gave). Show all of your work. (Hint: using the result of (b) will considerably narrow down the parameters of your search.)