integer factorization in Z(sqrt(d)) where Z(sqrt(d)) is a Euclidean ring - Maple Help

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numtheory[factorEQ] - integer factorization in Z(sqrt(d)) where Z(sqrt(d)) is a Euclidean ring

Calling Sequence

factorEQ(m, d)

Parameters

m

-

integer, list or set of integers in Zd

d

-

integer where d is a Euclidean ring

Description

• 

The factorEQ function returns the integer factorization of m in the Euclidean ring Zd.

• 

Given integers a and b of Zd, with b0, there is an integer q such that a&equals;bq&plus;r, normr<normb is true in Zd. In these circumstances we say that there is a Euclidean algorithm in Zd and that the ring is Euclidean.

• 

Euclidean quadratic number fields have been completely determined. They are Zd where d = -1, -2, -3, -7, -11, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, and 73.

• 

When d&equals;2,3mod4, all integers of Zd have the form a&plus;bd, where a and b are rational integers. When d&equals;1mod4, all integers of Zd are of the form 12a&plus;bd where a and b are rational integers and of the same parity.

• 

The answer is in the form: ±1uf1e1...fnen such that m=±1uf1e1fnen where f1,,fn are distinct prime factors of m, e1,,en are non-negative integer numbers, u is a unit in Zd. For real Euclidean quadratic rings, i.e.  d > 0, u is represented under the form wn or w&conjugate0;n or wn or w&conjugate0;n where w is the fundamental unit, and n is a positive integer.

• 

The expand function may be applied to cause the factors to be multiplied together again.

Examples

withnumtheory&colon;

factorEQ38477343&comma;11

3125&plus;3411125341185&plus;1611851611

(1)

expand

38477343

(2)

factorEQ3843433&comma;33

3323&plus;43352&plus;123352123311&plus;233258&plus;73358733

(3)

expand

3843433

(4)

factorEQ408294234124424229&comma;29

212&plus;1229121229521229411&plus;2294&plus;2938&plus;7299558726892&plus;331629325229

(5)

expand

408294234124424229

(6)

See Also

expand, GIfactor, ifactor, numtheory[sq2factor]


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