Primitive
Euler
Bricks   Fred
Curtis

Feb
2001 [Last
annotated
Dec
2004] 
My
interest
in
Euler
bricks
began
with
the
infamous
open
question:
is
there
is
an
Euler
brick
with
an
integer
body
diagonal?
The
results
here
came
from
patterns
observed
in
the
factorisation
of
edges
of
primitive
Euler
bricks.
The
derivations
here
are
original,
but
the
results
so
far
are
not
novel. 
[Synopsis]
[Working]
[Further directions]
[Chronology]
[References]
[Links]
Euler
bricks
are
bricks
with
all
edges
and
face
diagonals
integers,
named
after
one
of
their
first
investigators,
Leonhard
Euler
(17071783).
Each
pair
of
edges
forms
the
legs
of
a
Pythagorean
triple,
each
face
diagonal
the
hypotenuse
of
a
Pythagorean
triple.
Example
edges:
88, 234, 480.
The
term
Euler
brick
has
a
number
of
aliases

[Leech77]
uses
the
term
Classical
Rational
Cuboid.
[Rathbun1]
uses
the
term
Body
Cuboid
to
refer
to
an
Euler
brick
with
an
irrational
body
diagonal.
French
terms
include
brique
de
Pythagore
and
paralleloide
de
Pythagore.
Currently
[Dec
2001]
it
is
not
known
if
an
Euler
brick
can
have
an
integer
body
diagonal

this
question
also
has
a
number
of
aliases

the
Perfect
Box
Problem,
Perfect
Cuboid
Problem,
Rational
Box
Problem,
Rational
Cuboid
Problem,
Integer
Cuboid
Problem,
etc.
A
primitive
Euler
brick
is
an
Euler
brick
width
edges
x, y, z
with
the
constraint
gcd(x, y, z) = 1.
Example:
x = 44, y = 117, z = 240.
Some
properties
of
primitive
Euler
bricks:
Some
properties
of
(not
necessarily
primitive)
Euler
bricks:
These
properties
follow
directly
from
the
properties
of
primitive
bricks
by
considering
an
arbitrary
brick
as
a
(possibly
unit)
multiple
of
a
primitive
brick.
 At
most
one
edge
is
odd
 At
least
two
edges
are
divisible
by
3;
at
least
one
edge
is
divisible
by
9
 At
least
two
edges
are
divisible
by
4;
at
least
one
edge
is
divisible
by
16
 At
least
one
edge
is
divisible
by
5
 At
least
one
edge
is
divisible
by
11
Conjugate
Euler
bricks:
The
edges
of
any
Euler
brick
a, b, c
may
be
factored
to
show
the
underlying
primitive
Pythagorean
triples.
Let
k = gcd(a, b, c),
x = a / k,
y = b / k,
z = c / k,
so
x, y, z
is
a
primitive
Euler
brick. 
Let:  m_{z} = gcd(x, y)  r_{x}=x/m_{y}m_{z}  then:  x = m_{y}m_{z}r_{x}   [a, b] = k[x, y]  =  m_{z}k  [m_{y}r_{x}, m_{x}r_{y}] 
m_{y} = gcd(x, z)  r_{y}=y/m_{x}m_{z}  y = m_{x}m_{z}r_{y}   [a, c] = k[x, z]  =  m_{y}k  [m_{z}r_{x}, m_{x}r_{z}] 
m_{x} = gcd(y, z)  r_{z}=z/m_{x}m_{y}  z = m_{x}m_{y}r_{z}   [b, c] = k[y, z]  =  m_{x}k  [m_{z}r_{y}, m_{y}r_{z}] 
  Legs
of Pythagorean triples 
   Legs of primitive triples 


The
m_{i}
are
pairwise
relatively
prime
since
x, y, z
is
primitive;
the
r_{i}
are
pairwise
relatively
prime
(pairwise
common
factors
of
x, y, z
absorbed
into
the
m_{i});
gcd(m_{i}, r_{i}) = 1
since
x, y, z
is
primitive. When
primitive
Euler
bricks
were
factored
as
above
into
m_{i}
&
r_{i}
as
above,
patterns
emerged
in
which
bricks
appeared
in
pairs,
and
this
lead
to
the
observation:
substituting
m_{x}<=>r_{z},
m_{y}<=>r_{y},
m_{z}<=>r_{x}
we
obtain
the
conjugate
Euler
brick
a', b', c': 
 x' = r_{y}r_{x}m_{z}   [a', b'] = k[x', y']  =  r_{x}k  [m_{z}r_{y}, m_{y}r_{z}] 
y' = r_{z}r_{x}m_{y}   [a', c'] = k[x', z']  =  r_{y}k  [m_{z}r_{x}, m_{x}r_{z}] 
z' = r_{z}r_{y}m_{x}   [b', c'] = k[y', z']  =  r_{z}k  [m_{y}r_{x}, m_{x}r_{y}] 
  Different legs
of Pythagorean triples 
   Same
legs
of primitive
triples
as
conjugate 

To
be
written
up:
 No
other
bricks
are
obtained
by
other,
similar
substitutions
(symmetry
argument)
 An
Euler
brick
and
its
conjugate
are
distinct
(Lack
of)
Novelty
Though
the
derivations
on
this
page
are
original,
the
results
are
not
novel.
Exactly
two
edges
of
a
primitive
Euler
brick
are
divisible
by
3:
each
pair
of
edges
are
the
legs
of
some
pythagorean
triple;
at
least
one
leg
of
any
pythagorean
triple
is
divisible
by
3,
so
at
least
one
in
each
pair
of
edges
is
divisible
by
3,
i.e.
at
least
two
edges
are
divisible
by
3.
Not
all
edges
can
be
divisible
by
3
since
the
brick
is
primitive,
so
exactly
two
edges
are
divisible
by
3.
Further,
when
the
pair
of
brick
edges
divisible
by
3
are
divided
by
their
gcd,
the
quotients
are
legs
of
a
primitive
Pythagorean
triple,
so
one
quotient
is
further
divisible
by
3,
i.e.
at
least
one
edge
must
be
divisible
by
9. The
argument
in
the
previous
paragraph
also
holds
for
divisibility
by
4,
thus
at
exactly
two
edges
of
any
primitive
Euler
brick
are
divisible
by
4
and
at
least
one
edge
is
divisible
by
16. At
least
one
edge
of
any
(not
necessarily
primitive)
Euler
brick
is
divisible
by
5.
If
neither
leg
of
a
primitive
Pythagorean
triple
is
divisible
by
5
then
one
leg
is
±1 (mod 5)
and
the
other
is
±2
(mod 5)

see
Table
"Possible
values
of
a,b,c
modulo
5".
This
property
holds
when
the
legs
are
multiplied
by
any
number
nonzero
modulo
5,
so
it
applies
to
any
(not
necessarily
primitive)
Pythagorean
triple
with
neither
leg
divisible
by
5.
Suppose
an
Euler
brick
has
no
edge
divisible
by
5,
then
WLOG
some
edge
x=±1 (mod 5),
implying
the
other
edges
y
and
z
are
both
±2 (mod 5)
;
y
and
z
cannot
be
the
legs
of
a
Pythagorean
triple,
so
no
such
brick
exists,
i.e.
at
least
one
edge
of
any
Euler
brick
is
divisible
by
5. 
At
least
one
edge
of
any
(not
necessarily
primitive)
Euler
brick
is
divisible
by
11.
If
neither
leg
of
a
primitive
Pythagorean
triple
is
divisible
by
11
then
the
unordered
pair
of
leg
residues
(modulo
11)
is
one
of
(±1, ±2),
(±1, ±5),
(±2, ±4),
(±3, ±4),
(±3, ±5)

see
Table
"Possible
values
of
a,b,c
modulo
11".
This
set
of
unordered
pairs
is
closed
under
multiplication
by
any
number
nonzero
modulo
11,
so
the
property
applies
to
any
(not
necessarily
primitive)
Pythagorean
triple
with
neither
leg
divisible
by
11.
Choosing
the
edges
of
an
Euler
brick
such
that
no
edge
is
divisible
by
11
is
equivalent
to
finding
a
triangle
in
the
graph
in
figure
1;
no
triangle
exists,
so
at
least
one
edge
of
any
Euler
brick
is
divisible
by
11. 
Figure
1
 
 An
article
in
Google
News
Groups
cites
"Ivan
Korec"
as
the
author
of
a
number
of
papers
on
the
rational
cuboid
problem.
 There
are
assorted
parametric
forms
for
Euler
bricks,
but
as
far
as
I
know
no
group
of
parametric
forms
cover
all
possible
Euler
bricks
 Sept
2000

Derived
the
divisibility
results
&
existence
of
conjugate
Euler
bricks
 Oct
2000

First
draft
of
page
;
found
and
added
references
[Leech77],
[Mathworld],
[Rathbun1]
 Feb
2001

Added
reference
for
[Guy94]
;
made
introductory
prose
&
ramble
on
conjugates
(hopefully!)
more
readable
 Dec
2001

Added
reference
for
[Rathbun2]
 Nov
2002

Fixed
a
few
spelling
&
HTML
formatting
mistakes
 Dec
2004

Added
link
to
Bill
Butler's
page
 [Guy94]
Guy,
R.
K.

"Is
There
a
Perfect
Cuboid?''
§D18
in
Unsolved
Problems
in
Number
Theory,
2nd
ed.
New
York:
SpringerVerlag,
pp.
173181,
1994.
(The
bulk
of
this
material
is
an
abbreviated
version
of
[Leech77])  [Leech77]
Leech,
John

"The
Rational
Cuboid
Revisited"
American
Mathematical
Monthly
vol.
84,
p518533,
1977.
Erratum
in
vol.
85,
p472,
1978.
 [Mathworld]
Euler
Brick

mathworld.wolfram.com
 [Rathbun1]
Randall
L.
Rathbun

Computer
searches
for
the
Integer
Cuboid

an
update
 [Rathbun2]
Randall
L.
Rathbun

The
Rational
Cuboid
Table
of
Maurice
Kraitchik

http://arxiv.org/abs/math/0111229
(This
is
a
huge
list
of
the
12,517
euler
bricks
with
odd
side
less
than
2^{32})
 Bill
Butler
is
undertaking
a
search
for
solutions

to
date
[Dec
2004]
he
has
searched
up
to
7,000,000,000
for
the
odd
side
in
a
primitive
Euler
brick.