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Algorithms in Number Theory
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
24 April, 2011
(at
01:16
)
Iterated Lightningbolt
(
Euclidean algorithm
)
Fix integers
J
0
and
J
1
, and set
D
:= Gcd(
J
0
,J
1
). A
pair (
s,t
) of integers is
“
a
B´
ezout pair
for
J
0
,J
1
”
if
sJ
0
+
tJ
1
=
D .
1a
:
B´
ezout’s lemma
says:
There always exists a B´
ezout
pair.
(
Alternative term:
s
and
t
are
B´
ezout multipliers
.
)
A B´
ezout pair (
s,t
) is not unique; it is (
except in the
boring
J
0
=0=
J
1
case
) part of a oneparameter family
s
k
:=
s
+
h
k
·
J
1
D
i
and
t
k
:=
t

h
k
·
J
0
D
i
,
1b
:
of B´
ezout pairs (
s
k
,t
k
), for each
k
∈
Z
.
1c
:
Exercise.
Prove that (1b) describes
all
the B´
ezout
pairs for
J
0
,J
1
.
±
Gcd
of several integers.
Given a list of integers,
~
J
= (
J
0
,J
1
,...,J
L
), use
Gcd(
J
0
,J
1
,...,J
L
) or Gcd(
~
J
)
2a
:
to denote the greatest common divisor,
D
, of the list.
Our goal is to simultaneously compute
D
and B´
ezout
multipliers
~
s
:= (
s
0
,...,s
L
) such that
X
L
‘
=0
[
s
‘
·
J
‘
]
=
D .
2b
:
We’ll accomplish this with
L
applications of
LBolt
:
D
note
==== Gcd
±
...
Gcd
(
Gcd(
J
0
,J
1
)
,J
2
)
...,J
L
²
.
Algorithm:
From integers
~
J
= (
J
0
,J
1
,...,J
L

1
,J
L
),
set
C
:= Gcd(
J
0
,J
1
,...,J
L

1
)
and
D
:= Gcd(
J
0
,J
1
,...,J
L

1
,J
L
)
note
==== Gcd(
C,J
L
)
.
Apply
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This note was uploaded on 01/26/2012 for the course MHF 3202 taught by Professor Larson during the Fall '09 term at University of Florida.
 Fall '09
 LARSON

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