View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino) POLITECNICO DI TORINO Repository ISTITUZIONALE The biharmonic mean Original The biharmonic mean / Abrate, Marco; Barbero, Stefano; Cerruti, Umberto; Murru, Nadir. - In: MATHEMATICAL REPORTS. - ISSN 1582-3067. - 18(2016). Availability: This version is available at: 11583/2719316 since: 2018-12-03T18:02:52Z Publisher: Romanian Academy, Publishing House of the Romanian Academy Published DOI: Terms of use: openAccess This article is made available under terms and conditions as specified in the corresponding bibliographic description in the repository Publisher copyright (Article begins on next page) 04 August 2020 What do Biharmonic mean? Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru University of Turin, Department of Mathematics Via Carlo Alberto 10, Turin Italy [email protected], [email protected] [email protected], [email protected] Abstract In this paper we describe some well{known means and their proper- ties, focusing on the relationship with integer sequences. In particular, the harmonic numbers, deriving from the harmonic mean, motivate the definition of a new kind of mean that we call biharmonic mean. Biharmonic mean allows us to introduce the biharmonic numbers and to provide a new characterization of primes. Biharmonic numbers ap- pear to be very interesting for some properties of divisibility which have been studied in our paper by means of linear recurrent sequences and diophantine equations. 1 Introduction The need to explore Nature and establish from direct observation its rules, encouraged ancient thinkers in finding appropriate mathematical tools, able to extrapolate numerical data. One of the oldest method used to combine observations in order to give an unique approximate value is the arithmetic mean. It has been probably used for the first time in the third century B.C., by ancient Babylonian astronomers, to determine the positions of celestial bodies. The mathematical concept of arithmetic mean was first enhanced by the greek astronomer Hipparchus (190{120 B.C.) and some other greek mathematicians, following the Pythagoric ideals, have also introduced or 1 rigorously defined further kinds of means. For example Archytas (428{360 B.C.) named the harmonic mean and used it in his theory of music and in an algorithm for doubling the cube. His disciple Eudoxus (408{355 B.C.) introduced the contraharmonic mean while he was developing the studies on proportions. So in parallel with the practical use of many numerical means in various sciences, a deep exploration of their arithmetic and geometric prop- erties took place during the centuries. The book of Bullen [2] is a classical reference for a good survey about means and their history. In this paper we especially focus our attention to some interesting arith- metic aspects related to the most used means in many fields of Mathematics. We also define a new kind of mean, showing how it also allows us to give a new characterization of prime numbers. We start recalling the classical definitions and properties of the involved means. Definition 1. Let a1; a2; : : : ; at be t positive real numbers, then we define their • arithmetic mean a + a + ··· + a A(a ; a ; : : : ; a ) = 1 2 t ; (1) 1 2 t t • geometric mean p t G(a1; a2; : : : ; at) = a1a2 ··· at; (2) • harmonic mean 1 1 1 1 −1 H(a1; a2; : : : ; at) = + + ··· + ; (3) t a1 a2 at • contraharmonic mean 2 2 2 a1 + a2 + ··· + at C(a1; a2; : : : ; at) = : (4) a1 + a2 + ··· + at We have the well{known inequalities H(a1; a2; : : : ; at) ≤ G(a1; a2; : : : ; at) ≤ A(a1; a2; : : : ; at) ≤ C(a1; a2; : : : ; at) 2 and only if we consider two positive real numbers a and b, we always have the following beautiful relations: A(a; b) = A(H(a; b); C(a; b)); (5) G(a; b) = G(H(a; b); A(a; b)): (6) A very interesting question is to determine whether at least one of these equalities holds for some choice of more than two positive real numbers. In 1948 Oystein Ore, ([4], [5]), introduced the idea of harmonic number finding some related properties and a quite surprising answer to this question. He applied the four means to all the divisors of a positive integer n. If we indicate the set of divisors of n as D(n) = fd1; d2; : : : ; dtg and for the sake of simplicity we pose A(n) = A(d1; d2; : : : ; dt); G(n) = G(d1; d2; : : : ; dt); H(n) = H(d1; d2; : : : ; dt); C(n) = C(d1; d2; : : : ; dt); recalling that the divisor function is t X x σx(n) = di ; x 2 N (7) i=1 the first immediate result of Ore can be summarized in the following theorem. Theorem 1. σ (n) A(n) = 1 ; (8) σ0(n) p G(n) = n; (9) nσ (n) H(n) = 0 ; (10) σ1(n) σ (n) C(n) = 2 : (11) σ1(n) 3 Proof. We give a proof only for equalities (10) and (9) since (8) and (11) clearly arise from (1) and (4) respectively. From equation (3) clearly 1 1 1 1 −1 H(n) = + + ··· + t d1 d2 dt and the sum 1 + 1 +···+ 1 , when reduced to the least common denominator d1 d2 dt n, gives as numerator σ1(n) (the sum of all the divisors of n). Moreover, since t = σ0(n) (the number of all the divisors of n), relation (10) easily follows. Now, to prove equation (9), we start from (2) v uσ0(n) pt σ0(nu) Y G(n) = d1d2 ··· dt = t di i=1 distinguishing the two cases n = m2 or n 6= m2. When n 6= m2, n has an n σ0(n) even number of divisors, so we multiply di with for all i = 1;:::; , di 2 σ (n) σ0(n) 0 Q 2 2 finding i=1 di = n and we are done. On the other hand if n = m , n then t = σ0(n) is odd. As before, we multiply di with , but in this case we di can do this only for every di 6= m, finding σ0(n) σ0(n)−1 σ0(n)−1 Y 2 σ0(n) di = n 2 m = (m ) 2 m = m : i=1 Thus v uσ0(n) p p σ0(nu) Y σ0(n) σ (n) G(n) = t di = m 0 = m = n: i=1 A straightforward consequence of these results, observed by Ore in [4], shows how an equality similar to (6) holds if we take into account the elements of D(n). We have Corollary 1. For any positive integer n G(n) = G(H(n);A(n)): (12) 4 Proof. From what we stated in the previous theorem, we clearly obtain s nσ (n) σ (n) p G(H(n);A(n)) = pH(n) · A(n) = 0 1 = n = G(n): σ1(n) σ0(n) Before considering another fashinating question, we want to point out that equality (12) can be interpreted also as a formal identity. For example when n = p2q, where p and q are two primes, D(n) = f1; p; q; pq; p2; p2qg and we have from (12) G(1; p; q; pq; p2; p2q) = G(H(1; p; q; pq; p2; p2q); A(1; p; q; pq; p2; p2q)): (13) The astonishing fact is that this equality also holds if we substitute p and q withp any other couple of positive real numbers. For example if we use in (13) 2 and π instead of p and q respectively, we obtain the identity p p p p p p G(1; 2; π; 2 π; 2; 2π) = G(H(1; 2; π; 2 π; 2; 2π); A(1; 2; π; 2 π; 2; 2π)): which is not so immediate. In general, as we have previously observed, for a set of t positive distinct real numbers a1; : : : ; at randomly choosed, the equality G(a1; a2; : : : ; at) = G(H(a1; a2; : : : ; at); A(a1; a2; : : : ; at)); is false. The second question is pretty natural: when do the means applied to the divisors of an integer n also give a result which is an integer? The case of G(n) is not so interesting because G(n) is an integer if and only if n is a square. All the integers n such that A(n) is also an integer are the terms of the sequence A003601 in OEIS [6]: 1; 3; 5; 6; 7; 11; 13; 14; 15; 17; 19; 20; 21; 22; 23; 27; 29; 30; 31; 33; 35; 37;::: (14) These integers are the so{called arithmetic numbers. Moreover, every integer n giving an integer value for C(n) belongs to the sequence A020487: 1; 4; 9; 16; 20; 25; 36; 49; 50; 64; 81; 100; 117; 121; 144; 169; 180; 196; 200; 225;:::: Maybe the most important case is related to the harmonic mean. Ore provided the following definition. 5 Definition 2. A positive integer n is an harmonic (divisor) number (or Ore number) if H(n) is an integer. The first harmonic numbers are 1; 6; 28; 140; 270; 496; 672; 1638; 2970; 6200; 8128; 8190; 18600; 18620; 27846; 30240;::: and they form the sequence A001599: The corresponding values for H(n) are listed in A001600 1; 2; 3; 5; 6; 5; 8; 9; 11; 10; 7; 15; 15; 14; 17; 24;:::: Ore also proved that all perfect numbers are harmonic numbers. In the next section, moving from the beautiful properties shown before, we define a new kind of mean that we will call biharmonic mean. Starting from the biharmonic mean we will define the biharmonic numbers, which appear to be very interesting since they allow us to provide a new characterization for prime numbers.