Simultaneous and quadratic are both types of what
Quadratic irrational number Quadratic irrational number In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the set of rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their common denominator, a quadratic irrational is an irrational root of some quadratic equation whose coefficients are integers. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be
Kaplansky's theorem on quadratic forms Kaplansky's theorem on quadratic forms In mathematics, Kaplansky's theorem on quadratic forms is a result on simultaneous representation of primes by quadratic forms. It was proved in 2003 by Irving Kaplansky. Kaplansky's theorem states that a prime "p" congruent to 1 modulo 16 is representable by both or none of "x" + 32"y" and "x" + 64"y", whereas a prime "p" congruent to 9 modulo 16 is representable by exactly one of these quadratic forms. This is remarkable since the primes represented by each of these forms individually are "not" describable by congruence conditions. Kaplansky's proof uses the facts that