IntegerExponent

The largest $k$ with $b^k \mid n$ — the $b$-adic valuation of $n$ as an integer.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageIntegerExponent!↗

IntegerExponent(n, b)the multiplicity of $b$ in $n$

SageMathInteger(n).valuation(b)SymPymultiplicity(b, n)Juliavaluation(ZZ(n), ZZ(b))

IntegerExponent(n)base 10

SageMathInteger(n).valuation(10)

Typed signatures

Type	Package	Applies
(number, number?) -> integer | number	number-theory	always

Domain: Number theory

Details

• Integers only: a rational's $p$-adic valuation is AdicValuation's, over AdicNumeral.
• $n = 0$ has every power of $b$ as a divisor, so IntegerExponent(0, b) is PositiveInfinity.
• $b$ need not be prime — IntegerExponent(n, 6) is the largest $k$ with $6^k \mid n$, not the 2-adic or 3-adic valuation.

Plot

Examples

3 more kept as data, checked against the oracles

Basic #

$2000 = 5^3 \cdot 16$

live ↗

In$\mathrm{IntegerExponent}(2\,000, 5)$Out$3$

base 10 by default

live ↗

In$\mathrm{IntegerExponent}(2\,000)$Out$3$

trailing zeros in base 10

live ↗

In$\mathrm{IntegerExponent}(1\,230\,000)$Out$4$

live ↗

In$\mathrm{IntegerExponent}(2^{10}×3^4, 2)$Out$10$

1152 is $2^7\cdot 9$, so the exponent of 2 is 7

live ↗

In$\mathrm{IntegerExponent}(1\,152, 2)$Out$7$

$144=2^4\cdot 9$

live ↗

In$\mathrm{IntegerExponent}(144, 2)$Out$4$

Scope #

Composite base: $1728=2^6 3^3$, so $6^3$ divides it but $6^4$ does not

live ↗

In$\mathrm{IntegerExponent}(1\,728, 6)$Out$3$

Negative integers

live ↗

In$\mathrm{IntegerExponent}(-1\,000)$Out$3$

Threads over a list

live ↗

In$\mathrm{IntegerExponent}(\bigl\lbrack10, 100, 1\,000\bigr\rbrack)$Out$\bigl\lbrack1, 2, 3\bigr\rbrack$

Gaussian integers: $8i=i\cdot(-i)^3(1+i)^6$

live ↗

In$\mathrm{IntegerExponent}(8i, 1+i)$Out$6$

Applications #

The trailing zeros of $100!$

live ↗

In$\mathrm{IntegerExponent}(100!)$Out$24$

The 2-adic valuation of each integer, the ruler sequence

live ↗

In$\mathrm{IntegerExponent}(1..50, 2)$Out$\bigl\lbrack0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1\bigr\rbrack$

Properties #

Legendre's formula: $\lfloor100/7\rfloor+\lfloor100/49\rfloor=16$

live ↗

In$\mathrm{IntegerExponent}(100!, 7)$Out$16$

$2524=4\cdot 631$

live ↗

In$\mathrm{IntegerExponent}(2\,524, 2)$Out$2$

Possible issues #

live ↗

In$\mathrm{IntegerExponent}(0, 5)$Out$∞$

Base 1 has no well-defined $p$-adic valuation, so the call stays unevaluated

live ↗

In$\mathrm{IntegerExponent}(10, 1)$Out$\mathrm{IntegerExponent}(10, 1)$

About

Examples	19
Checked against	Wolfram Language 18 agree1 differSageMath 16 agree2 errorSymPy 9 agree3 errorJulia 9 agree3 error
Crosswalk	Wolfram LanguageSageMath ×2SymPyJulia

Implementation

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: AdicValuation, FactorInteger