quadint

Fast, integer-backed algebraic number types for exact arithmetic in imaginary quadratic integer rings.

• complexint: a Gaussian integer type that mirrors Python’s complex, but stores int components (no floating-point drift).
• QuadInt / QuadraticRing: a general quadratic-integer implementation for elements of the form
  $(a + b\sqrt{D}) / \mathrm{den}$ with $den ∈ {1,2}$.
  • By default, QuadraticRing(D) chooses den = 2 when D % 4 == 1, otherwise den = 1 (and you can override with QuadraticRing(D, den=1) / den=2 to work in a non-default order).
• eisensteinint: Eisenstein integers in the ω-basis (a + bω, where $ω = (-1 + \sqrt{-3})/2$).
• dualint: dual integers of the form a + bε where ε² = 0 and ε != 0.
• splitint: split-complex (hyperbolic) integers of the form a + bj where j² = 1 and j != 1.

Designed for discrete math, number theory tooling, and high-throughput exact computations (this project is built to compile cleanly with mypyc).

New helper methods on every quadratic integer value:

• x.content() — largest positive integer n such that x = n*y in the same ring.
• x.factor_detail() — structured factorization as Factorization(unit, primes).
• x.factor() — a plain {prime_like_factor: exponent} mapping whose product is exactly x.
• x.basis, x.basis_a, x.basis_b — public/user-facing basis coordinates, which may differ from the internal (a, b) numerator coordinates.

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Installation

python -m pip install quadint

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Quickstart (recommended): complexint

from quadint import complexint

a = complexint(1, 2)
b = complexint(3, 6)

c = a * b
print(c)          # "(-9+12j)"  (exact, integer-backed)
print(c.real)     # -9
print(c.imag)     # 12
print(type(c.real))  # <class 'int'>

print(abs(a))     # 1^2 + 2^2 = 5  (norm)

complexint is ideal when you want something that feels like complex, but with infinite-precision integer components.

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Quadratic integers: QuadraticRing

Create a ring instance for a chosen discriminant parameter D, then construct values in that ring:

from quadint import QuadraticRing

Q2 = QuadraticRing(-2)  # Z[√-2]

x = Q2(1, 2)           # (1 + 2*sqrt(-2))
y = Q2(3, 6)

print(x * y)           # "(-21+12*sqrt(-2))"
print(abs(x))          # norm: 1^2 - (-2)*2^2 = 9

Common operations include +, -, *, ** (non-negative powers), conjugate(), and abs() (the norm).

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Eisenstein integers: eisensteinint

from quadint.eisenstein import eisensteinint

z = eisensteinint(2, 3)   # 2 + 3ω
w = eisensteinint(1, -1)  # 1 - ω

print(z)
print(z * w)              # exact product in Z[ω]
print(abs(z))             # norm (integer)

Use real and omega to access the ω-basis components.

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Dual integers: dualint

from quadint import dualint

z = dualint(2, 3)   # 2 + 3ε
w = dualint(1, -1)  # 1 - ε

print(z)
print(z * w)              # (2+1ε)

Use real and dual (or epsilon) to access the ε-basis components.

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Split-complex integers: splitint

Split-complex (a.k.a. hyperbolic) integers behave like complexint, except the generator satisfies j² = 1 instead of j² = -1.

Unlike complex numbers, split-complex numbers have an indefinite norm and zero divisors (e.g. (1+j)*(1-j) == 0).

from quadint.split import splitint

z = splitint(1, 1)    # 1 + 1j
w = splitint(1, -1)   # 1 - 1j

print(z * w)          # 0j   (zero divisor behavior)

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Division & interoperability notes

• This package is primarily intended for exact, discrete arithmetic (+, -, *, **, conjugation, norms).
• Division helpers (divmod, //, %, /) are implemented for the finite set of norm-Euclidean quadratic rings at the default/maximal denominator, and also for dual (D=0) and split-complex integers (D=1).
  • New: division is also available in selected Euclidean-but-not-norm-Euclidean real quadratic maximal orders via a Harper-style method (weighted Euclidean score + local quotient search). This currently covers:
    • D=14,22,23,31,43,46,47,53,59,61,62,67,71,77,83,86,89,93,94,97
    • and D=69 via a dedicated Clark-style Euclidean function implementation.
    • Without cypari, Harper-style support is limited to the built-in hard-coded cases above (with D < 100); with cypari installed, additional admissible Harper-like cases may be discoverable.
• Factorization (factor / factor_detail) is currently implemented for:
  • complexint (D=-1, den=1),
  • QuadraticRing(-2, den=1),
  • eisensteinint (D=-3, den=2),
  • and the Heegner maximal orders for D=-7 and D=-11. Other rings may raise NotImplementedError.
• Floats and Python complex are accepted in some operations but are converted via int(...), which truncates toward zero. If you care about rationals, avoid mixing in float.

Example of truncation behavior:

from quadint import complexint

a = complexint(3, 6)

print(a / 3)     # "(1+2j)"
print(a / 3.5)   # "(1+2j)"  (3.5 -> 3 by int(...) conversion)

print(a + 1)     # "(4+6j)"
print(a + 1.5)   # "(4+6j)"  (1.5 -> 1)

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Ideals and class numbers

quadint can work with integral ideals of quadratic orders.

from quadint import QuadraticRing

O = QuadraticRing(-5)
I = O.ideal(3, O(1, 1))

assert not I.is_principal()
assert O.class_number == 2

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Basis-vector coordinates

QuadInt separates public basis coordinates from the internal numerator coordinates used by the arithmetic engine. Internally, every value is still stored as (a, b) numerators for $(a + b\sqrt{D}) / \mathrm{den}$

For most quadratic integer types, the public basis is the identity basis, so the coordinates you pass to the constructor are the same coordinates used internally. Subclasses can override that by defining conversion matrices:

• BASIS_TO_INTERNAL maps constructor/user coordinates (x, y) into internal numerator coordinates (a, b).
• INTERNAL_TO_BASIS and INTERNAL_TO_BASIS_DEN map internal numerator coordinates back to public basis coordinates.

This is mainly useful when the natural mathematical notation for a type is not the raw 1, √D basis. Eisenstein integers are the motivating example. Users write them as a + bω, where $ω = (-1 + \sqrt{-3})/2$, but the shared quadratic-integer engine stores values over QuadraticRing(-3) as $(a + b\sqrt{D}) / \mathrm{den}$.

So eisensteinint(x, y) converts from the public ω-basis to the internal numerator basis as:

x + yω = ((2x - y) + y√-3) / 2

Example:

from quadint.eisenstein import eisensteinint

z = eisensteinint(2, 3)

print(z)             # (2+3ω)
print(z.real)        # 2
print(z.omega)       # 3
print(z.basis)       # (2, 3)
print(tuple(z))      # (2, 3)

# Internal numerator coordinates are still available, but usually only useful
# for implementing rings/subclasses or debugging low-level arithmetic.
print(z.a, z.b, z.ring.den)  # 1 3 2

Prefer basis, basis_a, basis_b, and type-specific aliases such as real / omega when presenting values to users. Prefer the internal .a and .b fields only when implementing arithmetic, division, factorization, or another low-level ring operation.

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Sums of squares and quadratic-form decompositions: quadint.sums

quadint.sums provides small number-theory helpers for decomposing primes and integers into non-negative integer solutions of:

x^2 + d*y^2 = n

The default d=1 gives the classic sum-of-two-squares problem.

from quadint.sums import decompose_prime, decompose_number

print(decompose_prime(19889))
# (17, 140)

print(decompose_number(19890))
# {(69, 123), (57, 129), (3, 141), (87, 111)}

Use d to solve related forms:

from quadint.sums import decompose_prime, decompose_number

print(decompose_prime(19, d=3))
# (4, 1) because 4^2 + 3*1^2 == 19

print(decompose_number(12, d=3, no_trivial_solutions=False))
# {(0, 2), (3, 1)} because 0^2 + 3*2^2 == 12 and 3^2 + 3*1^2 == 12

decompose_prime(p, d=1, den=1)

Return a non-negative pair (x, y) for a prime-like input where:

x^2 + d*y^2 = den^2 * p

For normal public use, leave den=1. Passing den=2 is mainly useful when working with denominator-2 quadratic orders, where the returned pair is in numerator coordinates.

from quadint.sums import decompose_prime

print(decompose_prime(5))
# (1, 2)

print(decompose_prime(7, d=3))
# (2, 1)

print(decompose_prime(2, d=7, den=2))
# (1, 1) because 1^2 + 7*1^2 == 2^2 * 2

decompose_number(n, d=1, ...)

Return all canonical non-negative integer pairs (x, y) satisfying:

x^2 + d*y^2 = n

from quadint.sums import decompose_number

print(decompose_number(325, no_trivial_solutions=False))
# {(1, 18), (6, 17), (10, 15)}

decompose_number accepts either an integer or a precomputed factorization dictionary:

from quadint.sums import decompose_number

print(decompose_number({2: 1, 3: 2, 5: 1, 13: 1, 17: 1}))
# same result as decompose_number(19890)

Useful options:

• d=1 by default; use another positive integer for x^2 + d*y^2 = n.
• no_trivial_solutions=True by default; set it to False to include solutions with a zero coordinate and symmetric d=1 solutions such as (0, 2) for n=4.
• check_count=N returns an empty set early when the predicted number of solutions is below N.

Completeness is best-supported for the class-number-one Heegner values used by the package: d in {1, 2, 3, 7, 11, 19, 43, 67, 163}. Other d values may work, and results are still validated as true solutions, but completeness is not guaranteed.

Eisenstein norm decompositions: quadint.sums.eisenstein

There is also a small companion module for decomposing Eisenstein norms of the form:

a^2 - a*b + b^2 = n

This is mostly a fun helper built on the Eisenstein integer machinery rather than a central part of the package. It mirrors the main quadint.sums API:

from quadint.sums.eisenstein import decompose_prime, decompose_number

print(decompose_prime(7))
# (1, 3)  # because 1^2 - 1*3 + 3^2 == 7

print(decompose_number(91, no_trivial_solutions=False))
# returns canonical pairs (a, b) with a^2 - a*b + b^2 == 91

no_trivial_solutions=True filters the obvious square-like rays where a == 0, b == 0, or a == b. As with the rest of quadint.sums, a factorization dictionary may be passed instead of an integer.

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Minimal API overview

Constructors

• complexint(a: int = 0, b: int = 0)
• eisensteinint(a: int = 0, b: int = 0) where a + bω
• dualint(a: int = 0, b: int = 0)
• splitint(a: int = 0, b: int = 0)
• QuadraticRing(D: int = 0, den: int = None)
  • If den is omitted (None), it defaults to 2 when D % 4 == 1, otherwise 1.

Ring instance (QuadraticRing)

• Q(a: int = 0, b: int = 0) -> QuadInt (constructs using the ring’s internal basis)
• Q.from_ab(a: int, b: int) -> QuadInt (construct with user coords, respecting den)
• Q.from_obj(x) -> QuadInt (embed int/float, and complex only when D == -1)

Value type (QuadInt)

• x.conjugate()
• abs(x) (norm)
• x.units (finite torsion unit subgroup exposed as a tuple)
• x.content()
• x.factor_detail() (returns Factorization(unit, primes))
• x.factor() (returns plain dict[QuadInt, int])
• divmod(x, y), x // y, x % y (where supported)
• Iteration/indexing over the stored coefficients: list(x), x[0], x[1]

quadint.sums

• decompose_prime(p: int, d: int = 1, den: int = 1) -> tuple[int, int]
• decompose_number(n: int | dict[int, int], d: int = 1, check_count: int | None = None, *, limited_checks: bool = False, no_trivial_solutions: bool = True, warn: bool = True) -> set[tuple[int, int]]