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 ) <---> ( d2 - c2 | 2cd | d2 + c2 )
is a one-to-one correspondence between the set of all cid-pairs and the set of all pythagorean triples ( x | y | z ) with x2 + y2 = z2, 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:
(*)     n2 + 1 = r2 + s2.
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 = a2 + b2, and q = c2 + d2. Then n is even, p < q holds, and:
(**)     n2 + 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 = a2 + b2, and q = c2 + d2 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 = c2 + d2.
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 ( c1 | d1 ) and ( c2 | d2 ) be cid-pairs with 2 < q1 = c12 + d12 and 2 < q2 = c22 + d22 such that q1 and q2 are coprime. Then there are at least two different cid-pairs ( c | d ) with c2 + d2 = q1q2 =: q, arising from numbers
c1:, d1:, c2: and d2: in the following way: let ( ai | bi ) be the pair attached to ( ci | di ).
n1 := a1c1 + b1d1 and n2 := a2c2 + b2d2. For q = q1q2 there are exactly two different even numbers n with 1 < n < q and n = t1q1 ± n1 = t2q2 ± n2 (t1, t2 integers). one of these numbers n, then q divides n2 + 1. To ( n | q ) there belongs (Theorem 6) a cid-pair ( c | d ) with q = c2 + d2, and, in addition, either
( c / d ) = ( |c1d2 - d1c2| / c1c2 + d1d2 ) or { c , d } = { d1d2 - c1c2 , c1d2 + d1c2 } 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 = d2 - c2. The number r = bd - ac is even, with r < w, and w divides r2 - 1.
Conversely, for every pair ( r | w ) with odd w and even r, obeying 2 ≤ r ≤ w - 3, and w | r2 - 1, there exists a unique cid-pair ( c | d ) with w = d2 - c2 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 ).