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PELL NUMBER TRIPLES
ERNST M.COHN
National Aeronautics and Space Administration, Washington, D.C. 20546
Horadam [1] has shown that Pythagorean triples a r e Fibonacci-number triple s„ It has
now been found that Pythagorean triples a r e Pell-number triples as well.
The Diophantine solution for Pythagorean triples (x, y, z)
and z = p 2 + q 2 , where p > q.
is x = 2pq,
F o r primitive solutions (p,q) = 1, p and
y = p 2 - q2,
q are of dif-
ferent parity, x o r y s 0 (mod 3) [ 2 ] } x = 0 (mod 4), and all prime factors of
congruent to 1 modulo 4.
z
are
Since x ^ y, r e g a r d l e s s of primitivity, let
y - x = p 2 - q2 - 2pq = ±c
(1)
which is readily transformed into
p - q = N/2q2 ±
(2)
and
p + q = N/2p2 ± c
(3)
.
It may be noted in passing that all values of c for primitive triples a r e of the form 12d ± 1
and 12d ± 5, where d = 0, 1, 2, 3, • e e .
However, l e s s than fifty percent of numbers of
this form a r e possible values of c, because this representation by means of three Pell numb e r s includes all odd numbers not divisible by 3.
Two characteristic identities of the Pell-number sequence,
(4)
Pn+2 = 2Pn+1 + Pn
(P„ = 0,
Pl
= 1)
and
(Pn+1 + Pn)2 - 2P^+1 = (-l)n+1
(5)
were used [3] to prove that Pell numbers generate all values for
(x, y, z) when c = 1.
Multiplication of (5) by a2 shows that Pell numbers also generate all values for
2
when c = a ,
c = 4, q
r e g a r d l e s s of primitivity.
= 2P ; c = 9, q
Thus, when c = 1,
= 3P , etc.
when c = 2a 2 , obviously nonprimitive,
q
= P
and p
(x, y, z)
= P +^_;
Similarly, Pell numbers generate all (x, y, z)
When c = 2,
q
= P
- +P
and p
= Pn+2
+
P .-; c = 8, Mq = 2(P _ + P ); c = 18,5 Mq = 3(P ,- + P / ),
etc.
n+1
n
n+1
n
n
n+1
n s
All other Pythagorean triples are represented by generalized Pell n u m b e r s , similar to
Horadam 1 s generalized Fibonacci numbers [ 4 ] , in such a way that a pair of equations is
associated with each value of c8
403
404
PELL NUMBER TRIPLES
(6a)
or
*2n+l
d
= aP
n+1
2n+2
Oct. 1972
bP
and
(6b)
qM0
_ = bP ^ + aP
2n+1
n+1
n
^2n+2
where a > b. The value of p , associated with a given value of q, is obtained by replacing
n by (n + 1). It will be noted that the odd and even values form two distinct sequences.
Upon combining (2) and (3) with (6), we obtain
(7a)
(7b)
b ) P n + 1 + (a
2n+l
<Wl
=
(a
2n+2
V+2
=
(a + b ) P n + 1
+
+
b)P t i
a)P
(b
and
(8a)
P
2n+1
+
q
2n+l
(8b)
p
2n+2
+
q
2n+2
(3a
b)Pn+1
+
(a
b)P
(a + 3 b ) P n + 1 + (a + b ) P n
where the subscripts for p and q may be interchanged between (7a) and (7b) as well as b e tween (8a) and (8b) a s needed.
Since Pell numbers p r o p e r , and generalized Pell numbers for primitive solutions, a r e
alternately of different parity, and with p ± q odd for primitive solutions,
a ± b must be
odd in view of (7) and (8). All other possible values of a ± b also occur and give r i s e to nonprimitive triples.
Thus, all (x51 y, z) can be generated, and no impossible values occur.
Once obtained, all values a r e easily verified, and any oversight of a permissible value of c
becomes obvious by the absence of an expected pair (a,b).
But there appears to be no s y s -
tematic, analytical method of determining a priori either possible values of c o r their a s sociated pair o r p a i r s of (a,b), except for c = a2 and c •=. 2a 2 , where b••••= 0.
Following is a table of the first 33 values for c, a,
and b > 0. Values of c
giving
r i s e to primitive solutions a r e underlined.
£
a
b
£
a
b
c
a
b
7
2
1
56
6
2
£7
7
6
14
62
7
1
98**
9
1
103
8
3
3
1
17
3
2
23
4
28
31
6
1
63
68
3
6
4
112
8
4
4
2
11
6
5
113
9
2
4
3
73
7
2
119*** 8
5
34
5
1
11
8
1
119*** 10
1
41
5
2
82
7
3
124
8
6
46
5
3
8£
7
4
126
9
3
7
2
47
92
2
127
8
5
4
94
*This also has the solution 7F~7
**This also has the solution 7 ( P * + 1 + P n )
5
136
10
6
1
49*
[Continued on page 412. ]
"ifc|4;"
This is the first value with two p a i r s
of solutions.