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A GENERALIZED PYTHAGOREAN THEOREM
A. G. SHANNON
University of Papua and New Guinea, Boroko, T, P. N. G.
and
A, F. HORADAfV!
University of New England, Armidale, Australia
1. INTRODUCTION
The results in this paper arose from the efforts of the first-named author
to adapt Horadam f s Fibonacci number triples [3] to generate direction of
numbers in solid geometry for a multivariable calculus course.
This effort
was unsuccessful in that the equation obtained was true for quadratic d i o phantine equations in general, but it did not use any properties of the Fibonacci sequence. However, it did give rise to some results for higher order
sequences.
Horadam [2], [3], [4] has studied the properties of a generalized Fibonacci sequence defined by
(1)
H ^ = H J_- + H ,
n+2
n+1
n
(n > 1)
with Hi = p 9 H2 = p + q. One of the properties he found was that
(2)
(H
n H n + 3>
+
<2Hn+lHn+2)2 =
(2H
n+lHn+2
+
H
n)2
Which connects generalized Fibonacci numbers with Pythagorean triples,,
In the next section of this paper, an analogous result is obtained for generalized "Tribonacci" numbers* The theorem is then extended to general
linear difference equations of order r with unit coefficients.
2* TRIBONACCI NUMBER TRIPLES
The general Tribonacci series (see Feinberg [1]) is defined by
(3)
U
n+3 =
U
n+2
+ U
n+1
+ U
n>
307
<n *
«
308
A GENERALIZED PYTHAGOREAN THEOREM
[May
with initial values Ui, U2» U3 .
Theorem l e
(4)
(Un Un+4
^
2
2
+ (2(Un+1
^ )U n+3
^ + U n+2
^ )2
A1 + U n+2
l Q ) = (U n + 2(U n+1
AO )U n+3
Proof,
u2 M u ^ - l u . + u
n
±0 )
)2
n+1
n+2
2n+3 +
= Un+3
(Un+1
_,, + Un+2
^)2
- 2(U n+1
^ +U n+2
^ 0 )U n+3
^Q
and so
K
+ 2
<Un+l
+ U
n + 2> U n + 3 = K+3
+
<Un+l
+ U
n + 2> 2 •
This gives
(Un2 + 2(Un+1
^ + Un+2
^o) 2
l 0 )U n+3
<5)
U
=
n+3
+
= K+B
<Un+l
+ U
n + 2 )4 + 2 < U n + l
~ <Un+l
+ U
n+2)2>2
+
+ U
<2<Un+l
n + 2 )U n + 3> 2
+ U
n+2)Un+3)2 •
Now
(Un+3-<Un+l+Un+2)2>2
(6)
=
<Un+3 " <Un+l
+ U
n+2>
)2 +
<Un+3 + <Un+l
= u2 u2
n n+4
Substitution of (6) in (5) gives the result (4).
Theorem 2. All Pythagorean triples are Fibonacci triples.
Proof,
Put Uj = x - y5
U4 = x
For n = 1, Eq. (4) becomes
U2 = y,
and
U3 = 0. Then
U5 = x + y .
+ U
n+2 ) ) 2
1971]
A GENERALIZED PYTHAGOREAN THEOREM
309
(x2 - y 2 ) 2 + (2xy)2 = (x2 + y2 )2 .
For example, when x = 5 and y = 2, we get the triple 20, 2 1 , 29*
3. GENERALIZED PYTHAGOREAN THEOREM
Comparison of (4) with (2) suggests that for a general recurring sequence
("V } of order r where
r-1
V
W
V
n+r = E
<n * « •
n+i •
i=0
witn initial values V l9 V2? s • 8 , V r , there is a Pythagorean theorem of the
form
Theorem 3e
(V V
)2 + (2 V
(V
- V ) )2
1
l
n n+r+1 ;
n + r l n+r
n; ;
2
= (V + 2V
(V
- V ) )2
lv
1
n . n + r n+r
n; ;
(8)
For example, when r = 2, we get
<Vn W
+
(2Vn+2Vn+l)2 = K
+ 2V
n+2Vn+l)2
which agrees with (2). When r = 3, we get
<VnVn+4>2
+
<2Vn+3^+2+Vn+l))2
= ^
n
+
2(Vn+1+Vn+2)Vn+3)2
which agrees with (4).
Lemma 1.
<9>
2V
Proof.
n + r " V n + r + l = Vn
310
A GENERALIZED PYTHAGOREAN THEOREM
n+r
[May
n+r+1
= V
+ V
- V
n+r
n+r
n+r+1
= v(V
-,+V
+ 9 « - + V ^ + V ;) + V
n+r-1
n+r-2 0
n+1
n
n+r
-(V
v
n+r
+V
., + V
« + • • • + V ., )
n+r-1
n+r-2
n+l'
= V .
n
L e m m a 2.
(10)
2 V _, + V ^ ^ = 4 V X
n+r
n+r+1
n+r
Proof.
-V
n
T h i s r e d u c e s i m m e d i a t e l y to
n+r
n+r+1
n
'
which h a s j u s t been proved.
Proof of T h e o r e m 3.
v(2 V
v
n+r
F r o m L e m m a s 1 and 2 9 we have
- V
., ) (2 V
+ V
., ) = V v (4 V
n + r + 1 / v n+r
n+r+1'
n
n+r
- V ) 9,
V
which b e c o m e s
4V 2
- V2
. = V v(4V
- V );
n+r
n+r+1
n
n+r
n
T h i s can be r e a r r a n g e d a s
(11)
v
;
V2
. = V2 + 4 V
(V
- V ) .
v
n+r+1
n
n + r x n+r
V
On multiplication by V 2 and addition of (2V , (V , - V ) ) 2 to each side
J
^
n
n+r n+r
n
of (11), the r e s u l t in (8) follows.
F o r example 9 when r = 4 5 we get a " t e t r a n a c c i n s e r i e s [1] and (7)
becomes
v
v
n+r
= y^ v .
£-** n+i
i=0
1971]
A GENERALIZED PYTHAGOREAN THEOREM
with Vi = V2 = V3 = V4 = 1,
time, (8) becomes
(V
(12)
+
n
2(
say. Then
V5 = 4 ,
V6 = 7.
311
At the same
\+l+Vn+2+Vn+3>Vn+4)2
= < V n V n + 5 )2
+
<2<Vn+l
+ V
n+2
+ V
n+3)Vn+4)2 •
n = 1 gives the Pythagorean triple 79 24, 25*
If we call the type of triangle in (8) a recurrence triple, we get
Theorem 49 All Pythagorean triples are recurrence triples*
Proof.
Put
Vi = x - y,
in (7). Then V5 = x 9
V2 = y,
Vg = x + j 9
(x2 - y 2 )
4e
2
V3 = V4 = 0
and for n = 1, Eq* (8) becomes
2
+ (2xy)2 = (x2 + y2 ) .
CONCLUDING COMMENTS
The results in Theorem 3 can be used to produce various properties for
recurrence relations of different orders.
For instance! when r = 2 and
n = m - 1, we get
(13)
v
2
2
2
4 (H'm+1- Hm - 1- - Hm+1
^ ) = Hm
^ ,
- 1- - Hm+2
which, in conjunction with Eq* (11) of [2], gives
<14>
4 H
< L -H U
+
^m^ = HL-i - = U •
where e = p2 - pq - q2*
For a third-order relation, the property analogous to (13) is
(15)
4(Um+2Um_1 - U^+2) = U ^
- U^+3 .
This may provide a convenient method for the development of properties of
third and higher order recurrence relations, which have been studied in a
312
A GENERALIZED PYTHAGOREAN THEOREM
number of papers in the Fibonacci Quarterly in recent years.
May 1971
For earlier
studies, Morgan Ward [5] provides a useful reference.
REFERENCES
1. M. Feinberg, "Fibonacci-Tribonacci," Fibonacci Quarterly, Vol. 1, No.
3, 1963, pp. 71-74.
2. A. F. Horadam, "A Generalized Fibonacci Sequence," American Mathematical Monthly, Vol. 68, 1961, pp. 455-459.
3. A. F. Horadam, "Fibonacci Number T r i p l e s , " American Mathematical
Monthly, Vol. 68, 1961, pp. 751-753.
4. A. F. Horadam, "Complex Fibonacci Numbers and Fibonacci Quaternions,"
American Mathematical Monthly, Vol. 70, 1963, pp. 289-291.
5. M. Ward, "The Algebra of Recurring S e r i e s , " Annals of Mathematics,
Vol. 32, 1931, pp. 1-9.
[Continued from page 306„ ]
3. H. W. Gould, "Generating Functions for Products of Powers of Fibonacci
Numbers," Fibonacci Quarterly, Vol. 1, No. 2, 1963, pp. 1-16.
4. H. W. Gould, "Up-Down Permutations," SIAM Review, Vol. 10, 1968,
pp. 225-226.
5. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numb e r s , Oxford University P r e s s , 1965, Ch. XVII.
6. A. F. Horadam, "A Generalized Fibonacci Sequence," The American
Mathematical Monthly, Vol. 68, 1961, pp. 445-459.
7. E. Landau, "Sur la serie des inverses des nombresdeFibonacci, Bulletin de la Societe Mathematique de France, Vol. 27, 1899, pp. 298-300.