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ON A CONJECTURE OF PIERO FILIPPONI
Richard Andre-Jeannin
IUT GEA, Route de Romain, 54400 Longwy, France
{Submitted February 1992)
1. INTRODUCTION
Let us define a generalized Lucas sequence {Hn(m)} by
H„(m) = Hn^m)+mHn_2(m),
H0(m) = 2, Hx(m) = 1,
(1)
where m > 1 is a natural number.
In a communication that appeared in a recent issue of this journal [1], P. Filipponi showed
that
Hp,(p) = l (mod//)
(2)
wherep is an odd prime, and he proposed also the following Conjecture:
Hp,(p-l)
= l (mod//)
(3)
where p > 5 is a prime number.
Following a method introduced by Lucas ([2], p. 209; [3]), we shall prove here generalizations of (2) and (3), namely,
Theorem 1: If p > 1 is a natural number, and if AW = 0 (mod p), then
H M(m) = l (modps+ll
s>0.
Theorem 2: If p > 5 is a prime number and if m = -1 (mod/?), then
H s(m) = l (modps+l\
$>Q.
2e PRELIMINARIES
Let us recall Waring's formula
[p/2]
xp +yp
= (x + yy
+p^
(-l)kCp,k(xy)k(x
+ yy~2k,
where p is a natural integer, and
In our proofs, we shall need the following three lemmas.
Lemma 1: (i) If/? is a natural integer, then p,Cp^k is integral;
(ii) Ifp is a prime, then C
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k
is integral.
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ON A CONJECTURE OF PIERO FILIPPONI
Proof: (i) The result follows from the relation
^-M+fo-r1'
(ii) From the relation
* VV(P-4p-krl
and since gcd(&, p-k) = 1, it is clear that & divides f^J" 1 ) •
Lemma 2: If /? = ±1 (mod 6) is a natural number, then YJ-kp=i\-l)k Cp^k = 0.
Proof: Let us put x = eZ7r/3 and>> = e /7r/3 in Waring's formula to get
2cos/>;r/3 = l + / ? X ( - l ) * C / a ,
A:=l
and the conclusion follows from this, since 2 cospn 13 = 1, when/? = ±1 (mod 6).
Lemma 3: If/? is an odd integer, then (£p-l)p
=~l (modps+l), £>0.
Proof: The statement clearly holds for s = 0. Supposing that (£p-l)p
A is an integer, one can write
= -l + Aps+l, where
(^-i/+1=(-i+^+y
= (-l)p+(^\-l)p-lAps+l
+(P)(-l)p-2A2p2s+2
+ .-. + Appp(s+l) ^ -1
(modps+2\
since p is odd and ff] = /?.
Let us return to the recurrence relation (1). We have Hn(m) - anm+ finm where am and f}m
are the real numbers such that am + /3m - 1 and amf3m = -m. Following Lucas ([2], p. 212), we
replace x (resp. y) by apm (resp. f3pm ) in Waring's formula to get
Hp„1(m) = HP,(m) + pX
{-\)k{l+p)Cp,kmkp
Hp~2k(m\
(4)
where/? is a natural number.
3. PROOF OF THEOREM 1
The case/? = 1 needs no comment, since H1 = 1, so we suppose in the sequel that p > 2, and
thus that [p/2] >1.
Let us write Hn instead of Hn(m) in (4), to get
H•
x
=H> +(-\)l+r*pmrSH^+
[p/2]
£ {-\f{l+pS)PCp^Hp;2k,
(5)
k=2
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[FEB.
ON A CONJECTURE OF PIERO FILIPPONI
since CPtl = l. Notice that the last sum is empty foip = 2mdp = 3 and that pC k is an integer,
by Lemma l(i).
We proceed by induction upon s. The statement clearly holds for s = 0 since Hx = l.
Now, let us suppose that
ff,El
(modps+l).
By using an argument similar to the one used in Lemma 3, one can easily deduce from this that
j y j s i (modPMy
(6)
Next we have, for every s > 0 and every p>2, ps >2S >s + \ and thus
(a) pmpS = 0 (modps+2).
On the other hand we have, for every k > 2,kps > 22" = 2s+l > 5 + 2, and thus
(b) / ^ ' E O (modps+2).
Now, by using (6), (a), and (b) in (5), we have
(modps+2).
HpS+l=l
This concludes the proof of Theorem 1.
4. PROOF OF THEOREM 2
We suppose now that p>5 is a prime number, and thus that /? = ±l(mod6). Let us put
m = ^p - 1 in (4) and write Hn instead of Hn (£p -1) to obtain
[p/2]
H
, =/7£+/>£C„, f c (<p-l)*'ff;- 2 *.
(7)
We proceed by induction on 5. The statement clearly holds for s = 0, since Hx = l. Supposing that H s = 1 (mod/?*4"1), we obtain
Hp;2k = 1 (mod// +1 ), for 1 < k < [p/2],
(8)
and
Hpt=l
(modps+2y
(9)
On the other hand, we have, by Lemma 3,
(£p-l)kpS =(-l)k
(modps+l).
(10)
By Lemma l(ii), CPtk is an integer, and by (8), (10), and Lemma 2, we obtain
[p/2]
I,cp,k(tp-i)kpSHp;2k
k=l
[p/2]
= I c A , ( - i ) ^ o (modPs+ly
fc=i
Now, by (7), (9), and (11), it is clear thatif ,+1 =1 (modp
(ii)
). This concludes the proof of
Theorem 2.
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ON A CONJECTURE OF PIERO FILIPPONI
ACKNOWLEDGMENT
The author would like to thank the referee for his helpful and detailed comments.
REFERENCES
1. P. Filipponi. "A Note on a Class of Lucas Sequences." The Fibonacci Quarterly 29.3
(1991):256-63.
2. E. Lucas. "Theorie des fonctions numeriques simplement periodiques." Amer. J. Math. 1
(1878): 184-220, 289-321.
3. E. Lucas. The Theory of Simply Periodic Numerical Functions. Tr. from French by Sydney
Karavitz, ed. Douglas Lind. Santa Clara, Calif.: The Fibonacci Association, 1969.
AMS Classification Numbers: 11B39; 11B50
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