NumberTheory - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Group Theory : Numbers : NumberTheory/FactorNormEuclidean

NumberTheory

  

FactorNormEuclidean

  

factorization of integers in quadratic norm-Euclidean fields

 

Calling Sequence

Parameters

Returns

Description

Examples

Compatibility

Calling Sequence

FactorNormEuclidean(z, d, output_opt)

Parameters

z

-

integral element of Qd

d

-

rational integer such that Qd is a norm-Euclidean field

output_opt

-

(optional) equation of the form output = product or output = list; the default is output = product

Returns

• 

If output_opt is set to output = product, then the return value is of the form ±uap1b1pnbn where the pi are distinct prime factors and the bi are positive integers.

– 

If d>0, then u is either w or w&conjugate0; where w is the fundamental unit in Zd and a is a non-negative integer.

– 

If d<0, then u is a unit in Zd and a&equals;1.

• 

If output_opt is set to output = list, then the return value is of the form s&comma;x&comma;y&comma;a&comma;f1&comma;&comma;fn where s&equals;±1 and each fi is a three element list of the form p&comma;q&comma;k. Each p&plus;qd is a distinct prime and k is a positive integer.

– 

If d&gt;0, then u&equals;x&plus;yd where u is as previously described and a is a non-negative integer.

– 

If d<0, then x&plus;yd is a unit in Zd. Let t&equals;x&plus;yd. If t&equals;±1 then t&equals;s and x&comma;y&comma;a&equals;1&comma;0&comma;0. Otherwise, s&comma;a&equals;1&comma;1.

Description

• 

The FactorNormEuclidean function computes the integer factorization of z in the ring of integers Zd of the quadratic field Qd.

• 

Consider the absolute value of the field norm of Qd as a field extension of Q, denoted by N. If d is one of -11&comma;7&comma;3&comma;2&comma;1&comma;2&comma;3&comma;5&comma;6&comma;7&comma;11&comma;13&comma;17&comma;19&comma;21&comma;29&comma;33&comma;37&comma;41&comma;57&comma;73, then N satisfies the following property. If a and b are in Qd and b0, then there exists q and r in Qd such that a&equals;bq&plus;r and Nr<Nb. In this case, N is said to be a Euclidean function on Qd and Qd is said to be a norm-Euclidean field.

• 

When d&equals;2&comma;3mod4, integers in Zd have the form a&plus;bd and when d&equals;1mod4 they have the form a&plus;b12&plus;12d, where a and b are rational integers. Alternatively for when d&equals;1mod4, integers have the form a2&plus;b2d where a and b are rational integers of the same parity.

Examples

withNumberTheory&colon;

FactorNormEuclidean38477343&comma;11

385161185&plus;16111253411125&plus;3411

(1)

expand may be used to multiply together the terms.

expand

38477343

(2)

If output_opt option is explicitly set to output = product, the return value will be in product form.

FactorNormEuclidean3843433&comma;33&comma;output&equals;product

234333311&plus;23325873358&plus;73352123352&plus;1233

(3)

expand

3843433

(4)

If the output_opt is set to output = list, the return value will be in list form.

FactorNormEuclidean408294234124424229&comma;29&comma;output&equals;list

1&comma;52&comma;12&comma;4&comma;2&comma;0&comma;1&comma;4&comma;1&comma;0&comma;4&comma;1&comma;1&comma;11&comma;2&comma;0&comma;11&comma;2&comma;1&comma;38&comma;7&comma;0&comma;38&comma;7&comma;1&comma;12&comma;12&comma;1&comma;12&comma;12&comma;1&comma;9558726892&comma;3316293252&comma;0&comma;9558726892&comma;3316293252&comma;1

(5)

FactorNormEuclidean(z, d) returns an error if z is not an integer in Qd.

FactorNormEuclidean32&comma;2

Error, (in NumberTheory:-FactorNormEuclidean) 3/2 is not an integer in Q(sqrt(2))

Compatibility

• 

The NumberTheory[FactorNormEuclidean] command was introduced in Maple 2016.

• 

For more information on Maple 2016 changes, see Updates in Maple 2016.

See Also

expand

NumberTheory