Inlämningsuppgift 3

5.2 Determine all the odd primes that can be expressed in the form x^{2} + xy + 5y^{2}.

5.3 Determine all the positive integers that can be expressed in the form x^{2} + 2y^{2}.

5.4 Determine all the positive integers that can be expressed in the form x^{2} − y^{2}.

5.5 Show that there are precisely two reduced forms with discriminant −20. Hence prove that the primes that can be represented by x^{2} + 5y^{2} are 5 and those congruent to 1 or 9 (mod 20).

5.6 Calculate h(−31).

5.7 Find the least positive integer that can be represented by 4x^{2} + 17xy + 20y^{2}.

5.8 Prove that n and 2n, where n is any positive integer, have the same number of representations as the sum of two squares.