Giuga Numbers and the Arithmetic Derivative

José María Grau Departamento de Matematicas Universidad de Oviedo Avda. Calvo Sotelo, s/n 33007 Oviedo Spain

                        grau@uniovi.es

Antonio M. Oller-Marcén Centro Universitario de la Defensa Ctra. de Huesca, s/n 50090 Zaragoza Spain

                        oller@unizar.es

Abstract We characterize Giuga numbers as solutions to the equation n′ = an + 1, with a ∈ ℕ and n′ being the arithmetic derivative. Although this fact does not refute Lava’s conjecture, it does suggest doubts about its veracity.

1 Introduction

1.1 The Arithmetic Derivative

The arithmetic derivative was introduced by Barbeau [3] (see \underline{A003415} in Sloane’s On-Line Encyclopedia of Integer Sequences). The derivative of an integer is defined to be the unique map sending every prime to 1 and satisfying the Leibnitz rule, i.e., p′ = 1, for any prime p, and (nm)′ = nm′ + mn′ for any n, m ∈ ℕ. This map makes sense and is well-defined [7].

Proposition 1. If n = ∏_{i=1}^{k} p_{i}^{r_{i}} is the factorization of n in prime powers, then the only way to define n′ satisfying the desired properties is

n′ = n ∑_{i=1}^{k} \frac{r_{i}}{p_{i}} .

1.2 Giuga numbers

In [4], Giuga numbers were introduced in the following way motivated by previous work by Giuga [6].

Definition 2. A Giuga number is a composite number n such that p divides \frac{n}{p} − 1 for every prime divisor p of n.

It follows easily from the definition that every Giuga number is square-free.. There are several characterizations of Giuga numbers. We present the most important in the following proposition (note that we will consider ℕ = {1, 2, … }).

Proposition 3. Let n be a composite integer. Then, the following are equivalent:

i) n is a Giuga number.

ii) Giuga [6]: ∑_{p|n} \frac{1}{p} − ∏_{p|n} \frac{1}{p} ∈ ℕ.

iii) Borwein et al. [4]: ∑_{j=1}^{n−1} j^{ϕ(n)} ≡ −1 (mod n), where ϕ is Euler’s totient function.

iv) Agoh [1]: nB_{ϕ(n)} ≡ −1 (mod n), where B is a Bernoulli number.

To date, only thirteen Giuga numbers are known (see \underline{A007850} in Sloane’s On-Line Encyclopedia of Integer Sequences):

• With 3 factors:

30 = 2 · 3 · 5.

• With 4 factors:

858 = 2 · 3 · 11 · 13,
1722 = 2 · 3 · 7 · 41.

• With 5 factors:

66198 = 2 · 3 · 11 · 17 · 59.

• With 6 factors:

2214408306 = 2 · 3 · 11 · 23 · 31 · 47057,
24423128562 = 2 · 3 · 7 · 43 · 3041 · 4447.

• With 7 factors:

432749205173838 = 2 · 3 · 7 · 59 · 163 · 1381 · 775807,
14737133470010574 = 2 · 3 · 7 · 71 · 103 · 67213 · 713863,
550843391309130318 = 2 · 3 · 7 · 71 · 103 · 61559 · 29133437.

• With 8 factors:

244197000982499715087866346 = 2 · 3 · 11 · 23 · 31 · 47137 · 28282147·
3892535183,
554079914617070801288578559178 = 2 · 3 · 11 · 23 · 31 · 47059 · 2259696349·
110725121051,
1910667181420507984555759916338506 = 2 · 3 · 7 · 43 ⋯ 1831 · 138683 · 2861051·
1456230512169437.

There are no other Giuga numbers with fewer than 8 prime factors. There is another known Giuga number (found by Frederick Schneider in 2006) which has 10 prime factors, but it is not known if there is any Giuga number between this and the previous ones. This largest known Giuga number is the following:

4200017949707747062038711509670656632404195753751630609228764416 142557211582098432545190323474818 = 2 · 3 · 11 · 23 · 47059 · 2217342227· 1729101023519 · 8491659218261819498490029296021· 658254480569119734123541298976556403.

Observe that all known Giuga numbers are even. If an odd Giuga number exists, it must be the product of at least 14 primes. It is not even known if there are infinitely many Giuga numbers.

In 2009, Paolo P. Lava, an active collaborator of the On-Line Encyclopedia of Integer Sequences, conjectured that Giuga numbers were exactly the solutions of the differential equation n′ = n + 1, with n′ being the arithmetic derivative of n. It is not a well-known conjecture (see the book by Lava [2] or comments about sequence \underline{A007850} in Sloane’s On-Line Encyclopedia of Integer Sequences) but it seems known enough to appear in wikipedia’s article on Giuga numbers [8].

2 Giuga numbers and the arithmetic derivative: bringing doubts on Lava’s conjecture

It is surprising that Lava’s conjecture has not been resolved. Certainly the thirteen known Giuga numbers satisfy n′ = n + 1. Nevertheless, we can observe that this is exclusively due to the fact that these thirteen known Giuga numbers satisfy ∑_{p|n} \frac{1}{p} − ∏_{p|n} \frac{1}{p} = 1.

Let us introduce now a novel characterization of Giuga numbers in terms of the arithmetic derivative.

To do so we first need the following technical lemma [7, Corollary 2]..

Lemma 4. If n′ = an + 1, then n is square-free.

Proof. If there is a prime p such that p^{2} divides n, then p must divide n′. Since n′ = an + 1 this yields a contradiction.

Proposition 5. Let n be an integer. Then n is a Giuga number if and only if n′ = an + 1 for some a ∈ ℕ.

Proof. If n is a Giuga number then it is composite and square-free. Thus, we can put n = p_{1}p_{2} ⋯ p_{k} con k > 1 and since n is a Giuga number

∑_{i=1}^{k} \frac{1}{p_{i}} − ∏_{i=1}^{k} \frac{1}{p_{i}} = ∑_{i=1}^{k} \frac{1}{p_{i}} − \frac{1}{n} = a ∈ ℤ.

Consequently, n ∑_{i=1}^{k} \frac{1}{p_{i}} − 1 = an and it follows by definition that n′ = an + 1.

Conversely, assume that n′ = an + 1 with a ∈ ℕ. Then n is square-free, i.e., ∏_{p|n} \frac{1}{p} = \frac{1}{n}.

Thus, an + 1 = n′ = n ∑_{i=1}^{k} \frac{1}{p_{i}} implies that ∑_{i=1}^{k} \frac{1}{p_{i}} − \frac{1}{n} = a ∈ ℕ, i.e., n is a Giuga number, as

claimed.

This result shows that Lava’s conjecture is as close (or far away) to being refuted as the discovery of a Giuga number with

∑_{i=1}^{k} \frac{1}{p_{i}} − \frac{1}{n} > 1.

This is not likely to happen in the near future, since it is known that such a number must have more than 59 prime factors [5]. In any case it has been pointed out that Lava’s conjecture is not plausible and it could well be false.

References

[1] T. Agoh, On Giuga’s conjecture, Manuscripta Math. 87 (1995), 501–510.

[2] G. Balzarotti and P. Lava, 103 Curiosità Matematiche, Hoepli, Milan, 2010.

[3] E. J. Barbeau, Remarks on an arithmetic derivative, Canad. Math. Bull. 4 (1961), 117– 122.

[4] D. Borwein, J. M. Borwein, P. B. Borwein, and R. Girgensohn, Giuga’s conjecture on primality, Amer. Math. Monthly 103 (1996), 40–50.

[5] J. M. Borwein and E. Wong, A survey of results relating to Giuga’s conjecture on primality, in Advances in Mathematical Sciences: CRM’s 25 years (Montreal, PQ, 1994), CRM Proc. Lecture Notes, Vol. 11, Amer. Math. Soc., 1997, pp. 13–27.

[6] G. Giuga, Su una presumibile proprietá caratteristica dei numeri primi, Ist. Lombardo Sci. Lett. Rend. Cl. Sci. Mat. Nat. (3) 14 (1950), 511–528.

[7] V. Ufnarovski and B. Åhlander, How to differentiate a number, J. Integer Seq. 6 (2003), Article 03.3.4.

[8] Wikipedia, Giuga number, http://en.wikipedia.org/wiki/Giuga number.

2010 Mathematics Subject Classification: Primary 11B99; Secondary 11A99. Keywords: arithmetic derivative, Giuga number, Lava’s conjecture.

(Concerned with sequences \underline{A003415} and \underline{A007850}.)

Received March 11 2011; revised version received March 26 2012. Published in Journal of Integer Sequences, March 26 2012.

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