uniform distribution theory

Transcription

1 Uniform Distribution Theory no uniform distribution theory ON THE FIRST DIGITS OF THE FIBONACCI NUMBERS AND THEIR EULER FUNCTION Florian Luca Pantelimon Stănică ABSTRACT. Here we show that given any two finite strings of base b digits say and s 2 there are infinitely many Fibonacci numbers F n such that the base b representation of F n starts with and the base b representation of φf n starts with s 2. Communicated by Oto Strauch. Introduction Let b 2beandinteger.Let = c c kb be a positive integer written in base b. Washington [4] proved that there exist infinitely many Fibonacci numbers F n whose base b representation starts with. In fact the first digits of the Fibonacci sequence obey Benford s law in that the proportion of the positive integers n such that F n starts with is precisely log +/ /. Here we take this one step further. Let φm be the Euler function of the positive integer m. We put s 2 = d...d lb for some other positive integer written in base b and prove the following theorem. Theorem. Given positive integers = c c kb and s 2 = d...d lb written in base b there exist infinitely many positive integers n such that the base b representation of F n starts with the digits of and the base b representation of φf n starts with the digits of s 2. We use the fact that with α β = + 5/2 5/2 the Binet formula F n = αn β n holds for all n 0. α β For a positive real number x we write log x for the natural logarithm of x and x respectively {x} for the integer part respectively fractional part of x. 200 M a t h e m a t i c s Subject Classification: B39K36. K e y w o r d s: Euler function Fibonacci numbers digits distribution. 2

2 FLORIAN LUCA PANTELIMON STĂNICĂ 2. The proof By replacing with b m for some positive integer m if needed whose effect is adding m zeros at the end of the base b representation of we may assume that >s 2. By replacing s 2 by b m respectively s 2 b m for an arbitrary positive integer m we may assume that the length of the base b representation of thatisk is as large as we wish. In Section 4 of [2] it is shown that φf n /F n is dense in [0 ]. So we take ε 0 /5b 2k and choose a positive integer a such that + ε s 2 +2ε φ. Now we take any prime p> and look at p.sincep> it follows that p = Fap and the two factors and p / on the right above are coprime indeed the only common prime factor of these two numbers could be p which is not the case since p>. Any prime factor q of p / is a primitive prime factor of F dp for some divisor d of a. Recall that a prime number q is said to be a primitive prime factor of F n if q divides F n but does not divide any F m for m<n. One of the properties of primitive prime factors q of F n when n>5isthat q ±modn. In particular every prime factor q of p / is congruent to ± modp. Let q...q t be all the prime factors of p /.Then 2p t q q t p p α ap. Thus t = Op/ log p. Then 22 φp p = t φfa qi = φ = φ = φ i= exp t i= t exp O exp O q log p + O q i q 2 q q

3 FIRST DIGITS OF THE FIBONACCI NUMBERS AND THEIR EULER FUNCTION = φf a +O. log p It implies that if p>expκε where κ>0 is some absolute constant then φp +0.5ε s 2 +.5ε. 2 p We now follow Washington s argument [4] to prove that there exist infinitely many primes p such that the base b representation of p starts with. For this it is enough to show that p = b N + ζ ap for some integer 0 ζ ap b N. 3 Note that since q 2p it follows that if p is sufficiently large say p>b k then p cannot equal b N and in particular if in the above formula 3 we have ζ ap 0 then in fact ζ ap. The above formula 3 yields α ap = 5 b N + 5ζ ap + β ap = 5 b N + x ap. Since ζ ap it follows that 5ζap + β ap > 5ζap + β ap 5 > and 0 <x ap = 5s b. N So if x ap 0 /b k and p > b k is sufficiently large it then follows that ζ ap <b N which is what we want. Thus or ap log α =log 5 +N +log+x ap ap log α N log { 5 = log } 5 + log + x ap. 4 Observe that log 5 / is never an integer. Assume that k is sufficiently large such that { b k < log } 5. Then putting δ = log + /bk we see that a relation like 4 with x ap 0 /b k holds provided that { } { a log α log } { 5 log } 5 p + δ. 5 23

4 FLORIAN LUCA PANTELIMON STĂNICĂ The number γ = a log α/ is irrational. By a result of Vinogradov [3] the sequence of fractional parts {pγ} p prime is uniformly distributed. In particular containment 5 holds for a positive proportion of primes p and therefore certainly for infinitely many of them. So indeed relation 3 holds. Relation 2 now shows that φp =s 2 b N + θ where θ ζ ap +0.5ε +0.5εb N ζ ap +.5ε +.5ε b N. Since ε</5b 2k the above upper bound is b ζ ap +.5ε +.5ε b N < b N k b k + 0. b k < b N + 0.bk b 2k b N where the last inequality above is implied by 9 < bn b N which holds true for all b 2andN. This completes the proof of the theorem. 3. Comments It was shown in [] that with σm being the sum of divisors of the positive integer m theratioσf n /F n is dense in [. The present method now shows that there are infinitely many positive integers n such that the base b representation of F n starts with the digits of and the base b representation of σf n starts with the digits of s 2. Also one may replace the Fibonacci sequence F n in the above statements with some other sequence u n for which it has been proved that φu n /u n and u n /σu n respectively are dense in [0 ]. For example one can take u n =2 n see [] and the main result of this paper still holds provided that b is not a power of 2. We give no further details. Acknowledgments. We thank the referee for a careful reading of the paper and for comments which improved its quality. This paper was written during a visit of F. L. to the Applied Mathematics Department of Naval Postgraduate 24

5 FIRST DIGITS OF THE FIBONACCI NUMBERS AND THEIR EULER FUNCTION School in December 202. During the preparation of this paper F. L. was supported in part by Project PAPIIT IN0452 UNAM VSP N Department of the US Navy ONR Global and a Marcos Moshinsky fellowship. REFERENCES [] LUCA F.: On the sum of divisors of the Mersenne numbers Math. Slovaca [2] LUCA F. MEJÍA HUGUET V. J. NICOLAE F.: On the Euler function of Fibonacci numbers J. Integer Sequences A [3] VINOGRADOV I. M.: A new estimation of a trigonometric sum containing primes Bull. Acad. Sci. URSSSer. Math [4] WASHINGTON L. C.: Benford s law for Fibonacci and Lucas numbers Fibonacci Quart Received January Accepted June F. Luca Mathematical Institute UNAM Juriquilla Santiago de Querétaro México and School of Mathematics University of the Witwatersrand P. O. Box Wits 2050 SOUTH AFRICA fluca@matmor.unam.mx P. Stănică Naval Postgraduate School Applied Mathematics Department Monterey CA USA pstanica@nps.edu 25