Aliquot sums of Fibonacci numbers

Transcription

1 Aliquot sums of Fibonacci numbers Florian Luca Instituto de Matemáticas Universidad Nacional Autónoma de Méico C.P , Morelia, Michoacán, Méico Pantelimon Stănică Naval Postgraduate School Alied Mathematics Deartment Monterey, CA 93943, USA July 24, 2007 Abstract Here, we investigate the Fibonacci numbers whose sum of aliquot divisors is also a Fibonacci number (the rime Fibonacci numbers have this roerty). Introduction Let (F n ) n be the sequence of Fibonacci numbers. For a ositive integer n we write σ(n) for the sum of divisors function of n. Recall that a number n is called multily erfect if n σ(n). If σ(n) = 2n, then n is called erfect. In [2], it was shown that there are only finitely many multily erfect Fibonacci This aer was written while at Auburn University Montgomery, Deartment of Mathematics, Montgomery, AL , USA

2 numbers, and in [3], it was shown that no Fibonacci number is erfect. For a ositive integer n, the value ϕ(n) of the Euler function is defined to be the number of natural numbers less than or equal to n and corime to n. Let s(n) = σ(n) n. The number s(n) is sometimes called the sum of aliquot divisors of n. Two ositive integers m and n (with m n) are called amicable if s(m) = n and s(n) = m. It is not known if there eist infinitely many amicable airs, but Pomerance [5] showed that the sum of the recirocals of all the members of all amicable airs is convergent. Here, we search for Fibonacci numbers F n such that s(f n ) is a Fibonacci number. In articular, rime Fibonacci numbers have the above roerty. We ut A = {n : s(f n ) = F m for some ositive integer m}. In this aer, we give an uer bound on the counting function of A. Theorem. There eists a ositive constant c 0 such that the inequality #A() < c 0 log log log holds for all > e ee. Throughout this aer, we use the Vinogradov symbols, and the Landau symbols O, and o with their usual meanings. We recall that A B, B A and A = O(B) are all equivalent and mean that A < cb holds with some constant c, while A B means that both A B and B A hold. For a ositive real number we write log for the maimum between 2 and the natural logarithm of. Note that with this convention, the function log is sub-multilicative; i.e., the inequality log(y) log log y holds for all ositive numbers and y. For a ositive real number t and a subset B of the ositive integers, we write B(t) = B [, t]. We use, q, P and Q with or without subscrits to denote rime numbers. Acknowledgements. During the rearation of this aer, F. L. was suorted in art by rojects PAPIIT 04505, SEP-CONACyT and a Guggenheim Fellowshi. 2 The Proof of Theorem Let be a large ositive real number. 2

3 2. Some sieving Let ω(n) and Ω(n) be the number of rime divisors of n and the number of rime ower divisors of n (> ), resectively. Let A () = {n : ω(n) < 0.9 log log or Ω(n) >. log log }. () By the Turán-Kubilius inequalities (see [8]) (f(n) log log ) 2 = O( log log ) n for both f {ω, Ω}, we infer that Let y = (log log ) /3 and let By Brun s sieve, We now write #A () log log. (2) A 2 () = {n : n for all rimes < y}. (3) #A 2 () <y ( ) log y σ(f n ) = F n + F m, log log log. (4) and we look at bounds for m in terms of n, where n does not belong to A () A 2 (). 2.2 Bounds for m in terms of n We start with a lower bound for m. Let γ = (+ 5)/2 be the golden section. Let n not in A () A 2 (). Then, there eists < y such that n. Thus, F F n, therefore γ m > F m = s(f n ) F n F γ n γ n y, where we used the fact that F n γ n. Hence, m n y + O(), 3

4 therefore m n 2y, once is sufficiently large. We now look at an uer bound for m. Note that γ m n F m σ(f n) F n F n F n ϕ(f n ) ( + ). (5) For every rime number let z() be its order of aarition in the Fibonacci sequence, and for a ositive integer d let P d = { : z() = d}. It is known that ± (mod z()) holds for all rimes > 5 and it is clear that F d P d (d ) #Pd, therefore F n #P d d log d. (6) Furthermore, z() log. We now get by taking logarithms in (5) that m n F n + O() d n + O(). P d Obviously, P d ± (mod d) <d 2 + #P d d 2 2 log log d, ϕ(d) where in the above inequality we have used estimate (6) as well as the known fact that the inequality ( ) log log t (a, b) + O, (7) ϕ(b) a (mod b) <t holds uniformly in corime ositive integers a < b and ositive real numbers t, where (a, b) is the first rime in the arithmetic rogression a (mod b) 4

5 (see, for eamle, [4]). Since the function log log d is sub-multilicative, we get that m n ( ( µ )) log log( ν ) + O ν µ n ν= e O log log + log log( ν ) ν n 2 ν 2 = e O log log +. n Since n A (), it follows that ω(n) <. log log. Thus, if we write < 2 <... for the increasing sequence of all the rime numbers, then n log log ω(n) i= log log ω(n) 2 log log t dπ(t) t (log log ω(n) ) 2 (log log log log ) 2. Hence, m n e ( O((log log log log ) 2 ) ) < 2y, where the last inequality holds if is large. In conclusion, if n is not in A () A 2 (), then m [n 2y, n + 2y]. 2.3 More sieving Let Q = {q : z(q) < q /3 }. Note that uniformly in t >, 2 #Q(t) q F n < γ n<t /3 n < γ t2/3, q Q n<t /3 q<t therefore #Q(t) t 2/3, 5

6 which shows that q Q q>s q s #Q(t) t s + /3 t d#q(t) t= t=s s + s #Q(t) t 2 dt t. (8) 4/3 s/3 We now ut u = (log ) 3 and let A 3 () = {n, z() n for some > u}. (9) For every fied rime > u, the number of n which are multiles of z() is /z() /z(). So, #A 3 () z() >u >u Q u /3 <d u /3 <d u /3 <d ± (mod d) <d 3 log log d dϕ(d) z() + >u Q d + + u /3 u /3 (log log d) 2 d 2 + u /3 z() (log log ) 2 d + (log log )2, (0) 2 u/3 (log ) /3 d>u /3 where in the above estimates we used (8) with s = u /3, the fact that φ(d) d/ log log d for all d, as well as estimate (7) with b = d and a = and d, resectively. We finally ut ω u (n) for the number of rime factors u of n, v = 2 log log log and let A 4 () = {n : ω u (n) > v}. () 6

7 Again by Turán-Kubilius inequality, (ω u (n) log log u) 2 = O( log log u), n< and since log log u = ( + o()) log log log, we get easily that #A 4 () log log log. (2) From now on, we deal only with numbers n which are not in A () A 2 () A 3 () A 4 (). 2.4 The 2-adic order of σ(f n ) Let K = 0.8 log log. Since n 4 i=a i (), we get that n has ω(n) ω u (n) > 0.9 log log 2 log log log > K rime factors P > u, once is sufficiently large. Let P > P 2 >... > P K be the first (largest) rime factors of n. Then P K > u. Note that ( K ) F n/p...p F n = i F n/p...p K, i=0 F n/p...p i+ where by convention we take P 0 =. Let L i = F n/p...p i F n/p...p i+ for i = 0,..., K and L K = F n/p...p K. We net observe that L i and L j are corime for all 0 i < j K. Indeed, assume that i < j K and Q are such that Q gcd(l i, L j ). Then ( ) F n/p...p Q gcd F n/p...p i+, i. F n/p...p i+ However, it is well-known that the greatest common divisor aearing above divides P i+. Hence, Q = P i+, and Q F n, therefore z(q) n. Since Q > u > 5 for large, we get that Qz(Q) n contradicting the fact that n A 3 (). Thus, L i and L j are indeed corime for all i < j. In [6], Ribenboim and McDaniel studied square-classes of Fibonacci numbers. Given two integers m and n, they are in the same square-class if F m F n 7

8 is a square. It follows from their results that if m > 2n and n is sufficiently large, then m and n are not in the same square-class. In articular, if is large, then none of the numbers L i is a erfect square. Thus, there eists a rime Q i L i, such that the order at which Q i aears in L i (hence, in F n ) is odd. It is also clear that Q i is odd if is large enough (say if u > 3). Thus, K i= (Q i + ) is a divisor of σ(f n ), which roves that σ(f n ) is a multile of 2 K. 2.5 The conclusion Let A 5 () be the set of all ositive integers n A() which are not in 4 i=a i (). Let n < n 2 <... < n l be all the elements in A 5 (). Then there eists k i [ 2y, 2y] such that m i = n i + k i for all i =,..., l. Furthermore, 2 K σ(f ni ) = F ni + F ni +k i. Let M = 4y +. We show that if l > M, then n i+m n i is large whenever i l M. Indeed, let n i < n i+ <... < n i+m. Then k j [ 2y, 2y] for all j = i,..., i + M, and since there are at most 2 2y + < M + ossible values of k j and M + ossibilities for the inde j, it follows that there eist j < j 2 in {i,..., i + M} such that k j = k j2. Let k denote the common value of k j and k j2. Using the formula F n = (γ n δ n )/(γ δ), where δ = ( 5)/2 is the conjugate of γ, we note that the relation 2 K F nj + F nj +k gives γ n j ( + γ k ) δ n j ( + δ k ) 0 (mod 2 K ), (3) and similarly for n j2. Here and in what follows, we say that an algebraic integer α is a multile of an integer m if α/m is an algebraic integer. Write λ = n j2 n j. Then the above relation for n j2 gives γ n j γ λ ( + γ k ) δ n j δ λ ( + δ k ) 0 (mod 2 K ). (4) Multilying the congruence (3) by γ λ and subtracting it from congruence (4), we get that δ n j ( + δ k )(γ λ δ λ ) 0 (mod 2 K ). (5) Conjugating the above relation (5) and multilying the resulting congruences, we get + δ k + γ k γ λ δ λ 2 8

9 is an integer which is a multile of 2 2K. Noting that the above integer is nonzero, by taking logarithms we get λ log γ + O(M) 2K, therefore λ K. Thus, we just roved that n i+m n i K, therefore #A 5 () M K + M, (6) (log log ) 2/3 which together with the uer bounds (2), (4), (0) and (2) comletes the roof of Theorem. References [] N. L. Bassily, I. Kátai, and M. Wijsmuller, On the rime ower divisors of the iterates of the Euler-ϕ function, Publ. Math. (Debrecen) 55 (999), [2] F. Luca, Multily erfect numbers in Lucas sequences with odd arameters, Publ. Math. (Debrecen) 58 (200), [3] F. Luca, Perfect Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo (2) 49 (2000), [4] C. Pomerance, On the distribution of amicable numbers, J. Reine Angew. Math. 293/294 (977), [5] C. Pomerance, On the distribution of amicable numbers. II, J. Reine Angew. Math. 325 (98), [6] P. Ribenboim and W. L. McDaniel, Square classes in Lucas sequences having odd arameters, J. Number Theory 73 (998), [7] G. Tenenbaum, Introduction to analytic and robabilistic number theory, Cambridge U. Press, 995. [8] P. Turán, On a theorem of Hardy and Ramanujan, J. London Math. Soc. 9 (934),