ResidueClass

$k \bmod n$ as a VALUE — an element of $\mathbb{Z}/n\mathbb{Z}$ that arithmetic stays inside, after Sage's Mod(a, m). compute-engine's head, strict about moduli: IntegerMod is its Sage-flavoured sibling.

ResidueClass(k, n)the residue class of $k$, normalised into $[0, n)$

SageMathMod(k, n)

ResidueClass(u/v, n)a rational reads as $u \cdot v^{-1}$, when $v$ is a unit mod $n$

SageMathMod(u/v, n)

ResidueClass(ResidueClass(k, m), n)the same class read in $\mathbb{Z}/n\mathbb{Z}$, for $n \mid m$

SageMathMod(k, m)

Typed signatures

Type	Package	Applies
(any, any) -> value	compute-engine	always
(any, any) -> value	residuesreplaces compute-engine	always

Domain: Modular arithmetic

Details

• Mod answers an integer; ResidueClass IS the class, so +, ·, / and powers of it are computed in $\mathbb{Z}/n\mathbb{Z}$ — a negative power inverts, and dividing by a non-unit leaves the call standing
• A bare integer or rational next to a class is read in the same ring
• Classes of different moduli never combine: the sum stays as written, so $x + y - y = x$ holds. Sage's coercion to $\mathbb{Z}/\gcd(m, n)$ is not followed; IntegerMod follows it
• ChineseRemainder of classes is the class mod $\operatorname{lcm}$ that reduces to each, and MultiplicativeOrder of a unit is its order
• The elements of QuotientRing(Integers, n), $\mathbb{Z}/n\mathbb{Z}$
• Written $\overline{k}_{n}$; a \pmod{n} is read as a class too, a \bmod n is still Mod, and a \equiv b \pmod{n} is still a congruence
• A call that declines — dividing by a non-unit — stays unevaluated with a ResidueClass::ninv message, after Wolfram's PowerMod::ninv
• IntegerMod, the head's old name, is now the Sage-flavoured sibling: the same arithmetic, with classes of different moduli combined in $\mathbb{Z}/\gcd(m, n)$

Examples

10 more kept as data, checked against the oracles

Basic #

live ↗

In$\overline{10}_{7}$Out$\overline{3}_{7}$

$1/3$ is the inverse of 3: $3 \cdot 5 = 15 \equiv 1$

live ↗

In$\frac{1}{3}\pmod{7}$Out$\overline{5}_{7}$

Fermat: $3^6 \equiv 1 \pmod 7$

live ↗

In$(\overline{3}_{7})^6$Out$\overline{1}_{7}$

Scope #

live ↗

In$\frac{1}{\overline{3}_{7}}$Out$\overline{5}_{7}$

Applications #

live ↗

In$\mathrm{ChineseRemainder}(\overline{2}_{3}, \overline{3}_{5})$Out$\overline{8}_{15}$

live ↗

In$\mathrm{MultiplicativeOrder}(\overline{2}_{7})$Out$3$

Possible issues #

classes of different moduli do not combine: the sum stays as written

live ↗

In$\overline{2}_{4}+(\overline{1}_{6})$Out$\overline{2}_{4}+(\overline{1}_{6})$

2 is not a unit mod 4, and the message says so

live ↗

In$\frac{1}{\overline{2}_{4}}$Out$\frac{1}{\overline{2}_{4}}$

odd mod 4 and even mod 6 at once: no such class

live ↗

In$\mathrm{ChineseRemainder}(\overline{1}_{4}, \overline{2}_{6})$Out$\mathrm{ChineseRemainder}(\overline{1}_{4}, \overline{2}_{6})$

About

Examples	19
Crosswalk	SageMath

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram's Mod/PowerMod answer a plain Integer, not a persistent ring element -- the same distinction this head's own details draw against our own Mod. FiniteField/FiniteFieldElement DOES keep a residue as a persistent value with +, ·, ^-1 (kernel-verified: FiniteField[7][3]+FiniteField[7][5] = FiniteFieldElement[.., {1}], FiniteField[7][3]^-1 = {5}, matching this head's own examples) -- but only for prime modulus; FiniteField[4] silently reinterprets 4 as GF(2^2), a different structure with no zero divisors, not the ring Z/4Z this head also supports (kernel-verified wrong for our composite-modulus examples), and FiniteField[6] errors outright. No single template is safe across this head's mixed prime/composite examples, so it stays unmapped for wolfram.

mappedsagepackages/oracle/src/mappings.ts

See also: IntegerMod, QuotientRing, Mod, ChineseRemainder, AdicNumeral