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Fungrim entry: 0b4d4b

QD={(a,b,c):aZ1andbZandcZandb24ac=Dandbacand((b=aora=c)    (b0))andgcd ⁣(a,b,c)=1}\mathcal{Q}^{*}_{D} = \left\{ \left(a, b, c\right) : a \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, {b}^{2} - 4 a c = D \,\mathbin{\operatorname{and}}\, \left|b\right| \le a \le c \,\mathbin{\operatorname{and}}\, \left(\left(\left|b\right| = a \,\mathbin{\operatorname{or}}\, a = c\right) \implies \left(b \ge 0\right)\right) \,\mathbin{\operatorname{and}}\, \gcd\!\left(a, b, c\right) = 1 \right\}
Assumptions:D{3,4,}andDmod4{0,3}D \in \{-3, -4, \ldots\} \,\mathbin{\operatorname{and}}\, -D \bmod 4 \in \left\{0, 3\right\}
References:
  • H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, Definition 5.3.2
TeX:
\mathcal{Q}^{*}_{D} = \left\{ \left(a, b, c\right) : a \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, {b}^{2} - 4 a c = D \,\mathbin{\operatorname{and}}\, \left|b\right| \le a \le c \,\mathbin{\operatorname{and}}\, \left(\left(\left|b\right| = a \,\mathbin{\operatorname{or}}\, a = c\right) \implies \left(b \ge 0\right)\right) \,\mathbin{\operatorname{and}}\, \gcd\!\left(a, b, c\right) = 1 \right\}

D \in \{-3, -4, \ldots\} \,\mathbin{\operatorname{and}}\, -D \bmod 4 \in \left\{0, 3\right\}
Definitions:
Fungrim symbol Notation Short description
PrimitiveReducedPositiveIntegralBinaryQuadraticFormsQD\mathcal{Q}^{*}_{D} Primitive reduced positive integral binary quadratic forms
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ZZZ\mathbb{Z} Integers
Powab{a}^{b} Power
Absz\left|z\right| Absolute value
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("0b4d4b"),
    Formula(Equal(PrimitiveReducedPositiveIntegralBinaryQuadraticForms(D), SetBuilder(Tuple(a, b, c), Tuple(a, b, c), And(Element(a, ZZGreaterEqual(1)), Element(b, ZZ), Element(c, ZZ), Equal(Sub(Pow(b, 2), Mul(Mul(4, a), c)), D), LessEqual(Abs(b), a, c), Parentheses(Implies(Or(Equal(Abs(b), a), Equal(a, c)), GreaterEqual(b, 0))), Equal(GCD(a, b, c), 1))))),
    Variables(D),
    Assumptions(And(Element(D, ZZLessEqual(-3)), Element(Mod(Neg(D), 4), Set(0, 3)))),
    References("H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, Definition 5.3.2"))

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2019-08-17 11:32:46.829430 UTC