ProfiniteNumber

A profinite number: the rational $x + m\hat{\mathbb{Z}}$ in $\hat{\mathbb{Q}} = \hat{\mathbb{Z}} \otimes \mathbb{Q}$, known modulo $m$. Modulus 0 (the default) is just the rational $x$ itself.

ProfiniteNumber(x, m)$x$ known modulo $m$ — a coset of $m\hat{\mathbb{Z}}$

SageMathQhat(x, m)

ProfiniteNumber(x)exact: modulus 0, so just the rational $x$

SageMathQhat(x)

ProfiniteNumber(list)glue a list of AdicNumeral values by CRT into one profinite number

SageMathQhat(list)

Typed signatures

Type	Package	Applies
(list<value> | value, number?) -> value	adeles	always

Domain: Adèles and idèles

Details

• $\hat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z}$, the profinite completion of $\mathbb{Z}$; $\hat{\mathbb{Q}} = \hat{\mathbb{Z}} \otimes \mathbb{Q}$ is the finite adèle ring, one coordinate per prime bundled together.
• Add, Multiply, Negate, Divide, Power, Equal, NotEqual are wrapped to recognise a ProfiniteNumber operand, like AdicNumeral.
• Numerator and Denominator split a profinite number into its numerator and denominator as profinite numbers (the denominator as an ordinary integer); AdicNumeral(p, z) projects it to $\mathbb{Q}_p$; Fibonacci and LucasL take a profinite argument (Lenstra's profinite Fibonacci numbers).
• ProfiniteNumber({AdicNumeral(p1, …), AdicNumeral(p2, …), …}) reassembles a profinite number from its images at several primes by the Chinese remainder theorem.

Examples

24 more kept as data, checked against the oracles

Basic #

$12\hat{\mathbb{Z}} + 8\hat{\mathbb{Z}} = 4\hat{\mathbb{Z}}$

live ↗

In$\mathrm{ProfiniteNumber}(3, 12)+\mathrm{ProfiniteNumber}(5, 8)$Out$\mathrm{ProfiniteNumber}(0, 4)$

live ↗

In$\mathrm{ProfiniteNumber}(3, 12)\mathrm{ProfiniteNumber}(5, 8)$Out$\mathrm{ProfiniteNumber}(3, 12)$

equal cosets need not share a modulus

live ↗

In$\mathrm{ProfiniteNumber}(6, 20)=\mathrm{ProfiniteNumber}(6, 40)$Out$\mathrm{True}$

live ↗

In$\mathrm{Numerator}(\mathrm{ProfiniteNumber}(\frac{2}{3}, 5))$Out$\mathrm{ProfiniteNumber}(2, 15)$

live ↗

In$\mathrm{Denominator}(\mathrm{ProfiniteNumber}(\frac{2}{3}, 5))$Out$3$

projecting to $\mathbb{Q}_2$

live ↗

In$\mathrm{AdicNumeral}(2, \mathrm{ProfiniteNumber}(100, 24))$Out$\mathrm{AdicNumeral}(2, 4, 3)$

projecting to $\mathbb{Q}_3$

live ↗

In$\mathrm{AdicNumeral}(3, \mathrm{ProfiniteNumber}(100, 24))$Out$\mathrm{AdicNumeral}(3, 1, 1)$

live ↗

In$\mathrm{ProfiniteNumber}(26, 20)$Out$\mathrm{ProfiniteNumber}(6, 20)$

live ↗

In$\mathrm{ProfiniteNumber}(-1, 12)$Out$\mathrm{ProfiniteNumber}(11, 12)$

live ↗

In$\mathrm{ProfiniteNumber}(5, -12)$Out$\mathrm{ProfiniteNumber}(5, 12)$

live ↗

In$\mathrm{ProfiniteNumber}(\frac{7}{2}, \frac{3}{2})$Out$\mathrm{ProfiniteNumber}(\frac{1}{2}, \frac{3}{2})$

live ↗

In$\mathrm{ProfiniteNumber}(\frac{1}{3}, 0)$Out$\frac{1}{3}$

live ↗

In$\mathrm{ProfiniteNumber}(7, 30)-2$Out$\mathrm{ProfiniteNumber}(5, 30)$

live ↗

In$\mathrm{ProfiniteNumber}(3, 12)^2$Out$\mathrm{ProfiniteNumber}(9, 36)$

live ↗

In$\frac{\mathrm{ProfiniteNumber}(4, 12)}{2}$Out$\mathrm{ProfiniteNumber}(2, 6)$

Scope #

CRT: glued from its 2-adic and 3-adic images

live ↗

In$\mathrm{ProfiniteNumber}(\bigl\lbrack\mathrm{AdicNumeral}(2, 20, 5), \mathrm{AdicNumeral}(3, 7, 2)\bigr\rbrack)$Out$\mathrm{ProfiniteNumber}(52, 288)$

a rational value and a rational modulus

live ↗

In$\mathrm{ProfiniteNumber}(\frac{1}{2}, \frac{97}{5})+\mathrm{ProfiniteNumber}(\frac{1}{3}, 10)$Out$\mathrm{ProfiniteNumber}(\frac{1}{30}, \frac{1}{5})$

Applications #

Lenstra's profinite Fibonacci numbers

live ↗

In$\mathrm{Fibonacci}(\mathrm{ProfiniteNumber}(3, 10))$Out$\mathrm{ProfiniteNumber}(2, 11)$

Lenstra's profinite Lucas numbers

live ↗

In$\mathrm{LucasL}(\mathrm{ProfiniteNumber}(3, 10))$Out$\mathrm{ProfiniteNumber}(4, 11)$

Properties #

gluing the projections at every prime of the modulus is the identity

live ↗

In$\mathrm{ProfiniteNumber}(\bigl\lbrack\mathrm{AdicNumeral}(2, \mathrm{ProfiniteNumber}(100, 24)), \mathrm{AdicNumeral}(3, \mathrm{ProfiniteNumber}(100, 24))\bigr\rbrack)$Out$\mathrm{ProfiniteNumber}(4, 24)$

Possible issues #

5 does not divide the modulus, so nothing is known at 5

live ↗

In$\mathrm{AdicNumeral}(5, \mathrm{ProfiniteNumber}(100, 24))$Out$\mathrm{AdicNumeral}(5, \mathrm{ProfiniteNumber}(4, 24))$

only an exact divisor works; a profinite divisor leaves the call standing

live ↗

In$\frac{1}{\mathrm{ProfiniteNumber}(5, 12)}$Out$\frac{1}{\mathrm{ProfiniteNumber}(5, 12)}$

a non-integer profinite numerator leaves Fibonacci standing

live ↗

In$\mathrm{Fibonacci}(\mathrm{ProfiniteNumber}(\frac{1}{2}, 3))$Out$\mathrm{Fibonacci}(\mathrm{ProfiniteNumber}(\frac{1}{2}, 3))$

native behavior on a plain rational is untouched

live ↗

In$\mathrm{Numerator}(\frac{3}{4})$Out$3$

native behavior on a plain rational is untouched

live ↗

In$\mathrm{Denominator}(\frac{3}{4})$Out$4$

only ProfiniteNumber gets Numerator; Adele does not

live ↗

In$\mathrm{Numerator}(\mathrm{Adele}(5))$Out$\mathrm{Numerator}(\mathrm{Adele}(5))$

About

Examples	50
Checked against	SageMath 24 agree
Crosswalk	SageMath ×2

Implementation

mappedwolframpackages/oracle/src/mappings.ts

No adele/idele or profinite-integer type in Wolfram -- no prior art to map against. Wolfram Function Repository searched too; nothing for adeles or profinite integers.

mappedsagepackages/oracle/src/mappings.ts

Hertogh's adeles package (github.com/mathehertogh/adeles); the exact case, modulus 0. The list-of-AdicNumeral glue form also has arity 1, but AdicNumeral itself is unmapped for sage, so those examples simply fail to emit rather than being mis-evaluated by this row.

mappedsagepackages/oracle/src/mappings.ts

Hertogh's adeles package; $1 known modulo $2.

See also: Adele, Idele, AdicNumeral, ProfiniteDecomposition