Consider the array indicated below: 1. 1. 2. 5. 13. 34. 89. 1. 2. 4. 9. 22. 56. 145. 1. 3. 7 ... and let A -, m = 0, 1,
ADVANCED PROBLEMS AND SOLUTIONS Edited by RAYMOND E.WHITNEY Lock Haven State College, Lock Haven, Pennsylvania
Send all communications concerning Advanced Problems and Solutions to Raymond E. Whitney, Mathematics Department, Lock Haven State College, Lock Haven, Pennsylvania 17745.
This department especially welcomes problems believed to be new o r extending old
results.
P r o p o s e r s should submit solutions or other information that will a s s i s t the editor.
To facilitate their consideration, solutions should be submitted on separate signed sheets within two months after publication of the problems. Proposed by Verner £ Hoggatt, Jr., San Jose State College, San Jose, California
H-195
Consider the array indicated below: 1
1
1
2
2
4
1
1
5
9
3
4
13
22
34
56 16
7 11
38
89 145
27
1
1
5
6
65 16
22 1
1
(i) Show that the row sums are F„ , n ^ 2. (ii) Show that the rising diagonal sums are the convolution of (F2n-l^=0
and
{^2,2)};=0
,
the generalized numbers of H a r r i s and Styles. H-196
Proposed by J. B. Roberts, Reed College, Portland, Oregon.
(a) Let A0 be the set of integral p a r t s of the positive integral multiples of r,
1 + ^5 T
=
„
413
—
where
414
ADVANCED PROBLEMS AND SOLUTIONS and let A
-,
[Oct.
be the set of integral p a r t s of the numbers nr 2
m = 0, 1, 2, • • • ,
for n 61 A . Prove that the collection of Z
of all positive integers is the disjoint
union of the A.. J (b) Generalize the proposition in (a). H-197 Proposed by Lawrence Somer, University of Hiin o is, Urbana, Illinois.
Let { i r } _ n sion relationship:
be the t-Fibonacci sequences with positive entries satisfying the r e c u r -
t (t) u n
Vu n -.i
Z—i
i=l
Find u .lim n —•
oo
(t) n+1
u
n
SOLUTIONS HYPER -TENSION H-185 Proposed by L Carlitz, Duke University, Durham, North Carolina. Show that
(1 - 2x) n = £ (-Dn"k ( n k=0 ^ where
2Fi[a,b;
2
+ k k )
( 2 k k ) (1 - v ) n - k 2 F l [-k; n + k + 1; k + 1; x] / \ /
c; x] denotes the hypergeometric function.
Solution by the Proposer.
We s t a r t with the identity
E
(y - z) r z s
(2r + 3s)! r!s!(r + 2s)! ^
r,s=0 Now put y = u + v, z = v,
frv 1 ;
V
_
.2r+3s+l y*
+
1 - y - z
so that
( 2 r + 3s)L
JLJ r l s l ( r + 2s)l ,r.s=0 *
u +
U
v
= ,2r+3s+l v '
1 1 - u - 2v
1972]
ADVANCED PROBLEMS AND SOLUTIONS
415
The right-hand side of (*) is equal to
n=0 while the left-hand side oo
oo
E
r s V 1 / -x\kf2r + 3s + k \ , U V { 1} ^ ~ { k J(U k=0
(2r + 3s)l r l s l ( r + 2s)l
r,s=0
^
±
+ V)
It follows that
, +, 2.v ,n = \ " ^ ,u + ,v) ,k
L < k=0
. x k / 2 r + 3s + k \
( 1] n
'
\
k
(2r + 3s)l U vr
J rl8l(r
+
2s)l
s ,
,
r+s=n-k
E
,
ixn-r-s
r+s •
Now, l e t p = +2 (mod 5) and n - r = s - t. F r o m page 77 of the reference to Ruggles, we have F . ^ . - F.L. = ( - l ) j + 1 F . . i+J 13 i-3 for all i and j . Therefore F np+r ^ = Fr Fn --1 - Fr+1 , - Fn + Fr Fn
E
F
r
^- Fn - 2Fr+1 , - Fn + Fr F n - -1 + Fr+1 n
F
= F ^ - L F = (-l)r+1 F (mod p) F r+n r n ' n-r for all n and r . Thus ( - l ) r + t F _,. = ( - l ) r + t ( - l ) t + 1 F . = ( - l ) r + 1 F = F __ , v sp+t \ / \ / n-r np+r s_t
(mod p) . v
Hence U
np+r
= U
lFnP+r
+ U
B
"1)r+t
l sP+t
(
0 FnP+r-l
(U F
+ U
B U
l(-1)r+tFsP+t
0Fsp+t-l>
+
V-l)'-1*-1
- "
FIBONACCI IS A SQUARE H-187
Proposed by Ira Gessel, Harvard University, Cambridge, Massachusetts.
Problem: Show that a positive integer n i s a Fibonacci number if and only if either 2
5n + 4 o r 5n2 - 4 i s a square. Solution by the Proposer.
Let F 0 = 0, F 4 = 1, F
- = F + F _- be the Fibonacci s e r i e s and L 0 = 2, ^ = 1,
L ,- = L + L n be the Lucas s e r i e s . r+1 r r-1 (1)
It is well known that
(-I)" + F*r = F r + 1 F r _ 1
(2)
L r = F r + 1 + Fr_x Subtracting four times the first from the square of the second equation, we have L2
r
_ 4 ( - l ) r - 4F 2 = (F _,, - F - )2 = F 2 , r r+1 r-1 r
whence 5F 2 + 4 ( - l ) r = L 2 . r r
418
ADVANCED PROBLEMS AND SOLUTIONS
[Oct.
Thus if n is a Fibonacci number, either 5n2 + 4 o r 5n2 - 4 is a square. I have two proofs of the converse. F i r s t Proof.
We use the theorem (Hardy and Wright, An Introduction to the Theory of
Numbers, p. 153) that if p and q a r e integers, 2
l/2q ,
x is a real number, and |(p/q) - x |
