A cid-pair ( c | d ) consists of two integers with:
(1) c and d are relatively prime, (2) 0 ≤ c < d, (3) one of the numbers is even, the other is odd.
Theorem 1
( 2c - d | c ), if d < 2c
(case A)
The function
: ( c | d ) --> ( c* | d* ) =
( d - 2c | c ), if 2c < d < 3c
(case B)
( c | d - 2c ), if 3c < d
(case C)
maps any cid-pair ( c | d ) with 2 < d to a "reduced" cid-pair ( c* | d* ) with d*< d.
From the cid-pair ( 1 | 2 ) every other cid-pair ( c | d ) with 2 < d can be obtained in a unique way from a "sequence" of transformations A, B or C (the reverse mapping of R in each case) successively applied to ( 1 | 2 ), defined by:
: ( c | d ) --> ( d | 2d - c ),
: ( c | d ) --> ( d | 2d + c ) , and
: ( c | d ) --> ( c | 2c + d ).
Theorem 2
( c | d ) <---> ( d
^{2}
- c
^{2}
| 2cd | d
^{2}
+ c
^{2}
)
is a one-to-one correspondence between the set of all cid-pairs and the set of all
pythagorean triples
( x | y | z ) with x
^{2}
+ y
^{2}
= z
^{2}
, odd x, and coprime x,y.
Theorem 3
For every cid-pair ( c | d ) with 0 < c there exists one and only one "attached" cid-pair ( a | b ) with
a < b,c < d ,
and ad - bc = ±1. The pair ( a | b ) is obtained, when the sequence, belonging to ( c | d ) due to Theorem 1, is applied to ( 0 | 1 ) rather than to
( 1 | 2 )
.
Theorem 4
Let ( c | d ) be a cid-pair with 0 < c, and ( a | b ) its attached cid-pair with a < b,c < d , and
ad - bc = ±1.
Further let n = ac + bd, r = bd - ac, and s = ad + bc. Then n and r are even numbers, s is an odd number and:
(*) n
^{2}
+ 1 = r
^{2}
+ s
^{2}
.
Conversely, for any triple ( n | r | s ) of natural numbers with even n, even r, and odd s, for which (*) holds, there exists a unique cid-pair ( c | d ), such that, together with the attached cid-pair ( a | b ) the equations ad - bc = ±1, n = ac + bd, r = bd - ac, and s = ad + bc are valid. If
, the values of
a, b, c, and d
can be obtained from
n, r, and s
in the following way: Compute
h:=gcd(n-r|s-1)
and
k:=gcd(n-r|s+1)
,
then n-r = hk
:
2. If
, then
a=h
:
2,
b=(s+1)
:
k,
c=k
:
2,
d=(s-1)
:
h
holds; otherwise
implies
a=k
:
2,
b=(s-1)
:
h,
c=h
:
2,
and
d=(s+1)
:
k.
Theorem 5
Let ( c | d ) be a cid-pair with 0 < c, and ( a | b ) its attached cid-pair. Then ( b | d ) is a cid-pair too, with attached cid-pair ( a | c ), and the sequence belonging to ( b | d ) is the
of the sequence belonging to ( c | d ).
Theorem 6
Let ( c | d ) be a cid-pair with 0 < c, and ( a | b ) the attached cid-pair ( with
a < b,c < d ,
and
ad - bc = ±1 ).
Further let n = ac + bd, p = a
^{2}
+ b
^{2}
, and q = c
^{2}
+ d
^{2}
. Then n is even, p < q holds, and:
(**) n
^{2}
+ 1 = pq.
Conversely, for any triple ( n | p | q ) of natural numbers with even n and p < q, observing (**), there exists exactly one cid-pair ( c | d ), such that, together with the attached cid-pair ( a | b ) the equations
ad - bc = ±1,
n = ac + bd,
p = a
^{2}
+ b
^{2}
, and q = c
^{2}
+ d
^{2}
hold. The cid-pair ( c | d ) arises from numbers
and
as follows: The sequence of the
( n | q ) consists of odd many letters and is palindromic with central letter B. If every letter to the left of the middle letter B and the letter B itself are
, the sequence for ( c | d ) is established.
Theorem 7
Let q be a prime number of the form q = 4k + 1. Then there is exactly one cid-pair ( c | d ) with
q = c
^{2}
+ d
^{2}
.
The pair ( c | d ) arises from the number
as follows:
The function
i: x --> y,
where
( a | y )
means the cid-pair attached to
( x | q ),
has the property i(i(x))=x, and is a permutation of the 2k - 1 numbers
{
= 2, 4, ... q-3 }
. The map i has exactly one
z.
The sequence of the cid-pair ( z | q ) has odd many letters and is palindromic with central letter B. If every letter to the left of the middle letter B and the letter B itself are
, the sequence for ( c | d ) is established.
Theorem 8
Let ( c
_{1}
| d
_{1}
) and ( c
_{2}
| d
_{2}
) be cid-pairs with
2 < q
_{1}
= c
_{1}
^{2}
+ d
_{1}
^{2}
and
2 < q
_{2}
= c
_{2}
^{2}
+ d
_{2}
^{2}
such that q
_{1}
and q
_{2}
are coprime. Then there are at least two different cid-pairs ( c | d ) with
c
^{2}
+ d
^{2}
= q
_{1}
q
_{2}
=: q,
arising from numbers
c
_{1}
:
,
d
_{1}
:
,
c
_{2}
:
and
d
_{2}
:
in the following way: let ( a
_{i}
| b
_{i}
) be the pair attached to ( c
_{i}
| d
_{i}
).
n
_{1}
:= a
_{1}
c
_{1}
+ b
_{1}
d
_{1}
and
n
_{2}
:= a
_{2}
c
_{2}
+ b
_{2}
d
_{2}
.
For
q = q
_{1}
q
_{2}
there are exactly two different even numbers
n
with
1 < n < q
and
n = t
_{1}
q
_{1}
± n
_{1}
= t
_{2}
q
_{2}
± n
_{2}
(t
_{1}
, t
_{2}
integers).
one of these numbers n, then q divides
n
^{2}
+ 1.
To ( n | q ) there belongs (Theorem 6) a cid-pair
( c | d )
with
q = c
^{2}
+ d
^{2}
,
and, in addition, either
( c / d ) = ( |c
_{1}
d
_{2}
- d
_{1}
c
_{2}
| / c
_{1}
c
_{2}
+ d
_{1}
d
_{2}
)
or
{ c , d } = { d
_{1}
d
_{2}
- c
_{1}
c
_{2}
, c
_{1}
d
_{2}
+ d
_{1}
c
_{2}
}
holds.
Theorem 9
Let w be odd with 1 < w. Any factorization w = uv with coprime
u < v
is attached to the cid-pair
( c | d )
with
u = c - d, v = c + d,
w = d
^{2}
- c
^{2}
.
The number r = bd - ac is even, with r < w, and w divides r
^{2}
- 1.
Conversely, for every pair ( r | w ) with odd w and even r, obeying 2 ≤ r ≤ w - 3, and w | r
^{2}
- 1, there exists a unique cid-pair ( c | d ) with
w = d
^{2}
- c
^{2}
and r = bd - ac, which is obtained from
in the following way:
The mapping
i: x --> y,
where
( a | y )
denotes the cid-pair attached to
( x | w )
, is a bijection of the set
{
| 2 ≤ x ≤ w-3 and aw - xy = -1 }
with i(i(x)) = x. The number r is a
of this mapping i. The sequence of the cid-pair ( r | w ) is palindromic with central letter A. If every letter to the left of the middle letter A and the letter A itself are
, the sequence leading to ( c | d ) is established, together with the product representation
w = ( d - c )( d + c )
.