IntegerMod

$k \bmod n$ as a VALUE, after Sage's Mod(k, n): ResidueClass whose classes of different moduli combine in $\mathbb{Z}/\gcd(m, n)$.

IntegerMod(k, n)the class of $k$, normalised into $[0, n)$, as for ResidueClass(k, n)

SageMathMod(k, n)SymPyGF(n, symmetric=False)(k)

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

SageMathMod(u/v, n)SymPyGF(n, symmetric=False)(u/v)

Typed signatures

Type	Package	Applies
(any, any) -> residue	residues	always

Domain: Modular arithmetic

Details

• The Sage-flavoured sibling of ResidueClass, which is compute-engine's strict head and never combines classes of two moduli. IntegerMod is built on it: same normalisation, same ring arithmetic, and ChineseRemainder, MultiplicativeOrder and membership in QuotientRing(Integers, n) read it as the class it is
• Two classes meet in $\mathbb{Z}/\gcd(m, n)$, the largest ring both reduce to, and the answer lives there: $(2 \bmod 4) + (1 \bmod 6) = 1 \bmod 2$. This is Sage's coercion, and the divergence from compute-engine, which leaves that sum as written
• A bare integer next to a class is read in the class's ring, never the other way: $(2 \bmod 4) \cdot 5 = 2 \bmod 4$
• Equality and != compare in the common ring too, so IntegerMod(2, 4) == IntegerMod(0, 2) is True
• Moduli with $\gcd = 1$ have no ring but the trivial one, which Sage rejects; the call stays as written. A class mod 1 is the exception: it is 0 in the zero ring $\mathbb{Z}/1$, which every ring reduces to, so it absorbs any class beside it. Sage does the same, though a sum of three still raises when a coprime pair comes before the mod 1 class; ours does not depend on order Dividing by a class that is not a unit in the common ring stays unevaluated too, with a ResidueClass::ninv message
• A ResidueClass next to an IntegerMod never combines: pick one head, as the two behaviours differ, and the sum stays as written
• Mod and IntegerMod differ: Mod(17, 5) is the remainder $2$, a plain number, while IntegerMod(17, 5) is the class $2 \bmod 5$ and keeps its modulus. Sage's Mod(a, m) builds a class, so it is IntegerMod(a, m) here, and compute-engine's Mod follows Wolfram's
• Written and read as $(a\;\mathrm{mod}\;m)$ (and as (a mod m) in TraditionalForm), fenced whole so sums and products of classes stay unambiguous. a \pmod{n} reads as ResidueClass and a \bmod n as Mod, so type this form or the call to get an IntegerMod

Examples

20 more kept as data, checked against the oracles

Basic #

live ↗

In$({10}\;\mathrm{mod}\;7)$Out$(3\;\mathrm{mod}\;7)$

Scope #

classes of different moduli meet in $\mathbb{Z}/\gcd(4, 6)$: the sum is 1 mod 2, as in Sage

live ↗

In$(2\;\mathrm{mod}\;4)+(1\;\mathrm{mod}\;6)$Out$(1\;\mathrm{mod}\;2)$

live ↗

In$(2\;\mathrm{mod}\;4)(1\;\mathrm{mod}\;6)$Out$(0\;\mathrm{mod}\;2)$

2 mod 4 and 0 mod 2 agree in $\mathbb{Z}/2$

live ↗

In$(2\;\mathrm{mod}\;4)=(0\;\mathrm{mod}\;2)$Out$\mathrm{True}$

the integer reads in $\mathbb{Z}/7$, so the answer keeps the modulus

live ↗

In$5(3\;\mathrm{mod}\;7)$Out$(1\;\mathrm{mod}\;7)$

$3^{100} \bmod 7$, reduced without forming the integer

live ↗

In$((3\;\mathrm{mod}\;7))^{100}$Out$(4\;\mathrm{mod}\;7)$

every step stays in $\mathbb{Z}/7$: $((5 + 4)\cdot 3 + 6)^3 = 6$

live ↗

In$(((5\;\mathrm{mod}\;7)+(4\;\mathrm{mod}\;7))(3\;\mathrm{mod}\;7)+(6\;\mathrm{mod}\;7))^3$Out$(6\;\mathrm{mod}\;7)$

Mod gives the remainder, a plain integer; IntegerMod(17, 5) is the class and keeps its modulus

live ↗

In$17\bmod5$Out$2$

Simplify folds the classes of a sum into one, in $\mathbb{Z}/\gcd(4, 6)$

live ↗

In$\mathrm{Simplify}((2\;\mathrm{mod}\;4)+(1\;\mathrm{mod}\;6))$Out$(1\;\mathrm{mod}\;2)$

the classes fold and the symbolic term stays

live ↗

In$\mathrm{Simplify}(x+(2\;\mathrm{mod}\;4)+(1\;\mathrm{mod}\;4))$Out$x+(3\;\mathrm{mod}\;4)$

Possible issues #

gcd(4, 7) = 1: Sage rejects the sum, and so does this

live ↗

In$(3\;\mathrm{mod}\;4)+(5\;\mathrm{mod}\;7)$Out$(3\;\mathrm{mod}\;4)+(5\;\mathrm{mod}\;7)$

the divisor is 0 in $\mathbb{Z}/2$, not a unit

live ↗

In$\frac{(1\;\mathrm{mod}\;6)}{(2\;\mathrm{mod}\;4)}$Out$\frac{(1\;\mathrm{mod}\;6)}{(2\;\mathrm{mod}\;4)}$

the strict head leaves it unevaluated; IntegerMod combines the same classes in $\mathbb{Z}/2$ (see above)

live ↗

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

every integer is 0 in the zero ring $\mathbb{Z}/1$

live ↗

In$(3\;\mathrm{mod}\;1)$Out$(0\;\mathrm{mod}\;1)$

2 is no unit mod 4: the call stays, with a ResidueClass::ninv message

live ↗

In$\frac{1}{(2\;\mathrm{mod}\;4)}$Out$\frac{1}{(2\;\mathrm{mod}\;4)}$

About

Examples	35
Checked against	SageMath 25 agree2 differ8 errorSymPy 19 agree8 differ4 unevaluated1 errorWolfram Language 1 agreempmath 1 agreeJulia 1 agreeRust 1 agree
Crosswalk	SageMathSymPy

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram has no residue value that follows Sage's coercion: Mod/PowerMod answer a plain Integer, and FiniteField[n] keeps a residue as a persistent value but only for prime n (FiniteField[4] is GF(2^2), not Z/4Z), with no coercion between fields.

mappedsagepackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

SymPy's Mod is the plain remainder; the residue value is the element of GF(n), which for a composite n is Z/n. A class of one modulus does not combine with one of another, where Sage's coercion takes them to Z/gcd. Power and Divide over a residue skip their S() wrap (emit.ts), which would lift a class to its integer.

See also: ResidueClass, QuotientRing, Mod, ChineseRemainder