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

SymPymultiplicity(b, n)

IntegerExponent(n)base 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.

Examples

3 more kept as data, checked against the oracles

Basic #

$2000 = 5^3 \cdot 16$

In

\mathrm{IntegerExponent}(2\,000, 5)

Out

3

base 10 by default

In

\mathrm{IntegerExponent}(2\,000)

Out

3

trailing zeros in base 10

In

\mathrm{IntegerExponent}(1\,230\,000)

Out

4

In

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

Out

10

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

In

\mathrm{IntegerExponent}(1\,152, 2)

Out

7

$144=2^4\cdot 9$

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

In

\mathrm{IntegerExponent}(1\,728, 6)

Out

3

Negative integers

In

\mathrm{IntegerExponent}(-1\,000)

Out

3

Threads over a list

In

\mathrm{IntegerExponent}(\bigl\lbrack10, 100, 1\,000\bigr\rbrack)

Out

\bigl\lbrack1, 2, 3\bigr\rbrack

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

In

\mathrm{IntegerExponent}(8i, 1+i)

Out

6

Applications #

The trailing zeros of $100!$

In

\mathrm{IntegerExponent}(100!)

Out

24

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

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: $\lfloor100/7\rfloor+\lfloor100/49\rfloor=16$

In

\mathrm{IntegerExponent}(100!, 7)

Out

16

$2524=4\cdot 631$

In

\mathrm{IntegerExponent}(2\,524, 2)

Out

2

Possible issues #

In

\mathrm{IntegerExponent}(0, 5)

Out

∞

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

In

\mathrm{IntegerExponent}(10, 1)

Out

\mathrm{IntegerExponent}(10, 1)

About

Examples	19
Checked against	Wolfram Language 18 agree1 differSymPy 9 agree3 error
Crosswalk	Wolfram LanguageSymPy

Implementation

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

See also: AdicValuation, FactorInteger