Odd perfect numbers of a special form^{∗}^{†}

Tomohiro Yamada

Abstract

We shall show that there is a effectively computable upper bound of odd perfect numbers whose Euler factors are powers of fixed exponent.

1 Introduction

We denote by σ(N) the sum of divisors of N. N is said to be perfect if σ(N) = 2N. Though it is not known whether or not an odd perfect number exists, many conditions which must be satisfied by such a number are known. Suppose N is an odd perfect number. Euler has shown that N = p^{α}q_{1}^{2β_{1}} ⋯ q_{t}^{2β_{t}} for distinct odd primes p, q_{1}, ⋯, q_{t} with p ≡ α ≡ 1 (mod 4). Steuerwald[12] proved that we cannot have β_{1} = ⋯ = β_{t} = 1. McDaniel[8] proved that we cannot have β_{1} ≡ ⋯ ≡ β_{t} ≡ 1 (mod 3). If β_{1} = ⋯ = β_{t} = β, then it is known that β ≠ 2(Kanold[6]), β ≠ 3(Hagis and McDaniel[5]), β ≠ 5, 12, 17, 24, 62 (McDaniel and Hagis[9]), and β ≠ 6, 8, 11, 14, 18 (Cohen and Williams[2]). In their paper [5], Hagis and McDaniel conjecture that β_{1} = ⋯ = β_{t} = β does not occur. We cannot prove this conjecture. But we can prove that for any fixed β, all of the odd perfect numbers N can be effectively determined. Our result is as follows.

Theorem 1.1. Let β ≥ 1. If N = p^{α}q_{1}^{2β} ⋯ q_{t}^{2β} is an odd perfect number, then

ω(N) ≤ 4β^{2} + 2β + 3.  (1)

and

N ≤ 2^{4^{4β^{2}+2β+3}}.  (2)

∗2000 Mathematics Subject Classification: 11A05, 11A25.

†Key words and phrases: Odd perfect numbers.

2 Lemmas

Let us denote by v_{p}(n) the solution e of p^{e}||n. For distinct primes p and q, we denote by o_{q}(p) the exponent of p mod q and we define a_{q}(p) = v_{q}(p^{d} − 1), where d = o_{q}(p). Clearly o_{q}(p) divides q − 1 and a_{q}(p) is a positive integer. Now we quote some elementary properties of v_{q}(σ(p^{x})). Lemmas 2.1 and 2.2 are well-known. Lemma 2.1 follows from Theorems 94 and 95 in Nagell[10]. Lemma 2.2 has been proved by Zsigmondy[13] and rediscovered by many authors such as Dickson[3] and Kanold[7].

Lemma 2.1. Let p, q be distinct primes with q ≠ 2 and c be a positive integer. If p ≡ 1 (mod q), then v_{q}(σ(p^{c})) = v_{q}(c + 1). Moreover, if p ≢ 1 (mod q), then v_{q}(σ(p^{c})) = a_{q}(p)+v_{q}(c+1) if o_{q}(p) | (c+1) and v_{q}(σ(p^{c})) = 0 otherwise.

Lemma 2.2. If a > b ≥ 1 are coprime integers, then a^{n} − b^{n} has a prime factor which does not divide a^{m} − b^{m} for any m < n, unless (a, b, n) = (2, 1, 6) or a − b = n = 1, or n = 2 and a + b is a power of 2.

By Lemma 2.2, we obtain the following lemmas.

Lemma 2.3. Suppose p is a prime and n is a positive integer. If d | (n+1), d > 1 and (p, d) satisfies neither (p, d) = (2, 6) nor (p, d) = (2^{m} − 1, 2) for some integer m, then there exists a prime q with o_{q}(p) = d and q | σ(p^{n}).

Proof. We can apply Lemma 2.2 with (a, b) = (p, 1) and we see that there exists a prime p such that o_{p}(q) = d. q does not divide p − 1 since o_{p}(q) = d > 1. On the other hand, q divides p^{n+1} − 1 since 1 < d | (n + 1). Hence q | σ(p^{n}).

Lemma 2.4. If p is a prime and n is a positive integer, then ω(σ(p^{n})) ≥ τ(n + 1) − 1 unless p = 2 and n ≡ 5 (mod 6), or p is a Mersenne prime and n is odd.

Proof. If d | (n + 1) and d ≠ 1, then (p, n, d) satisfies the condition of Lemma 2.3. Hence σ(p^{n}) has a prime factor q_{d} satisfying o_{p}(q_{d}) = d. Thus we have ω(σ(p^{n})) ≥ #{d : d | (n + 1), d > 1} = d(n + 1) − 1.

The following lemma is Lemma 2 of Danpat, Hunsucker and Pomerance [4].

Lemma 2.5. If p, q are distinct primes with q ≠ 2 satisfying σ(q^{x}) = p^{y} for some positive integers x, y, then p ≡ 1 (mod q) or a_{q}(p) = 1.

By this lemma and Lemma 2.1, we immediately deduce the following result.

Lemma 2.6. Let p, q be distinct primes satisfying the condition of Lemma 2.5 and q^{f} | σ(p^{e}). Then q^{f−1}o_{q}(p) divides e + 1.

Proof. By Lemma 2.5, p ≡ 1 (mod q) or a_{q}(p) = 1. If p ≡ 1 (mod q), then q^{f} | σ(p^{e}) implies q^{f} = q^{f}o_{q}(p) | (e + 1) by Lemma 2.1. If a_{q}(p) = 1, then o_{q}(p) | (e + 1) and v_{q}(e + 1) ≥ f − a_{q}(p) = f − 1 by Lemma 2.1. In both cases, q^{f−1}o_{q}(p) divides e + 1.

Moreover, we quote a result in Kanold[6].

Lemma 2.7. Let N = p^{α}q_{1}^{2β_{1}} ⋯ q_{r}^{2β_{r}} be an odd perfect number and l be a common divisor of 2β_{1} + 1, ⋯, 2β_{r} + 1. Then l^{4} | N. Moreover, if l is a power of a prime q, then p ≠ q.

3 Proof of the first part of Theorem 1.1

Let N = p^{α}q_{1}^{2β} ⋯ q_{r}^{2β} be an odd perfect number.

First assume that 2β + 1 = l^{γ}, where l is a prime and γ is a positive integer. Various results referred in the Introduction of this paper allows us to assume that β ≥ 8 without loss of generality.

By Lemma 2.7, p ≠ l and l^{4γ} divides N. Hence l = q_{i_{0}} for some i_{0} and v_{l}(N) = 2β. We divide q_{1}, ⋯, q_{r} into four disjoint sets. Let

S = {i : q_{i} ≡ 1 (mod l)},
T = {i : q_{i} ≢ 1 (mod l), i ≠ i_{0}, q_{j} | σ(q_{i}^{2β}) for some 1 ≤ j ≤ r},

and

U = {i : q_{i} ≢ 1 (mod l), i ≠ i_{0}, q_{j} ∤ σ(q_{i}^{2β}) for any 1 ≤ j ≤ r}.

Then i ∈ S ∪ T ∪ U ∪ {i_{0}} and thus we have

r ≤ #S + #T + #U + 1.  (3)

Lemma 3.1.

#S ≤ 2β.

Proof. For i ∈ S, we have l | σ(q_{i}^{2β}) by Lemma 2.1. Hence

#S ≤ v_{l}(∏_{i∈S} σ(q_{i}^{2β})) ≤ v_{l}(2N) = v_{l}(N) = 2β.

Lemma 3.2.

#T ≤ (2β)^{2}.

Proof. If i ∈ T, then q_{j} | σ(q_{i}^{2β}) for some j ∈ S. Hence we have ∑_{j∈S} v_{q_{j}}(σ(q_{i}^{2β})) ≥ 1 for i ∈ T. By Lemma 3.1 we have

\begin{aligned}#T & ≤ ∑_{i∈T} ∑_{j∈S} v_{q_{j}}(σ(q_{i}^{2β})) = ∑_{j∈S} v_{q_{j}}(∏_{i∈T} σ(q_{i}^{2β})) \\ & ≤ ∑_{j∈S} v_{q_{j}}(2N) = ∑_{j∈S} v_{q_{j}}(N) ≤ (2β)^{2}. \end{aligned}

Lemma 3.3.

#U ≤ 1.

Proof. If i ∈ U and q is a prime dividing σ(q_{i}^{2β}), then q = p since q | 2N, σ(q_{i}^{2β}) is odd, and q ≠ q_{j} for any j. Thus σ(q_{i}^{2β}) = p^{ζ_{i}} for some positive integer ζ_{i}.

We shall show q_{i}^{2β} | σ(p^{α}). If q_{i} divides σ(q_{j}^{2β}) for some j, then q_{i} divides 2β + 1 = l^{γ} or 1 < o_{q_{i}}(q_{j}) | (2β + 1) = l^{γ} by Lemma 2.1. The former case cannot occur since q_{i} ≠ q_{i_{0}} = l. If 1 < o_{q_{i}}(q_{j}) | (2β + 1) = l^{γ}, then o_{q_{i}}(p) = l^{t} for some integer t > 0 and therefore q_{i} ≡ 1 (mod l), which is inconsistent with the assumption q_{i} ∈ U. Since q_{i}^{2β} | N | σ(N), we conclude that q_{i}^{2β} | σ(p^{α}).

We can apply Lemma 2.6 with (q, f, e) = (q_{i}, 2β, α) and deduce that q_{i}^{2β−1} divides (α + 1)/2 since q_{i} is odd and α ≡ 1 (mod 4). Hence (α + 1)/2 must be divisible by ∏_{i∈U} q_{i}^{2β−1} and therefore d(α + 1) ≥ 2(2β)^{#U}. By Lemma 2.4, we have ω(σ(p^{α})) ≥ 2(2β)^{#U} − 1.

On the other hand, we have ω(σ(p^{α})) ≤ ω(N) + 1 since σ(p^{α}) | 2N. Thus, from Lemmas 3.1 and 3.2 we obtain

2(2β)^{#U} − 1 ≤ ω(N) + 1 ≤ #S + #T + #U + 3 ≤ #U + (2β)^{2} + 2β + 3.

Since #U ≤ (2β)^{#U−1} ≤ (2β)^{#U}/16 by the assumption that β ≥ 8, we have

\frac{31}{16}(2β)^{#U} ≤ 2(2β)^{#U} − #U ≤ (2β)^{2} + 2β + 4 ≤ \frac{21}{16}(2β)^{2},  (4)

and therefore #U ≤ 1.

By Lemmas 3.1, 3.2, 3.3 and (3), we have ω(N) ≤ r + 1 ≤ 4β^{2} + 2β + 3, which is the desired result.

Next we assume that 2β + 1 = l_{1}^{γ_{1}}l_{2}^{γ_{2}} ⋯ l_{s}^{γ_{s}} with s ≥ 2, where l_{1}, ⋯, l_{s} are distinct primes. By Lemma 2.7, l_{i}^{4γ_{i}} divides N for each i. This clearly implies that there are at least s − 1 primes in l_{i}’s each of which is equal to q_{j} for some j. Hence we may assume that l_{i} = q_{i} for i = 1, ⋯, s − 1.

Let S = {i : q_{i} ≡ 1 (mod l_{1})}. As in the prime-power case, we derive that #S ≤ 2β. By Lemma 2.4, each σ(q_{j}^{2β}) has at least one prime factor q with o_{q}(q_{j}) = d for any d > 1 dividing 2β +1. If we denote by w the number of divisors of 2β + 1 divisible by l_{1}, then w = γ_{1}(γ_{2} + 1) ⋯ (γ_{s} + 1) ≥ 2^{s−1}. Thus each σ(q_{j}^{2β}) has at least 2^{s−1} − 1 prime factors ≡ 1 (mod l_{1}) and different from p, namely, belonging to S.

Hence we conclude that r ≤ 2β#S/(2^{s−1} − 1) ≤ (2β)^{2} and therefore ω(N) ≤ 4β^{2} + 1, which is more than we desired. This completes the proof of the first part of Theorem 1.1.

To obtain the second part of our theorem it remains to apply the result of Nielsen[11], who has shown that M ≤ (d + 1)^{4^{l}} for any positive integer n, d, M, l satisfying σ(M)/M = n/d and ω(M) = l.

References

[1] A. S. Bang, Taltheoretiske Undersøgelser, Tidsskrift Math. 5 IV (1886), 70–80 and 130–137.

[2] Graeme L. Cohen and Williams, R. J., Extensions of some results concerning odd perfect numbers, Fibonacci Quart. 23 (1985), 70–76.

[3] L. E. Dickson, On the cyclotomic function, Amer. Math. Monthly 12 (1905), 86–89.

[4] G. G. Dandapat, J. L. Hunsucker and Carl Pomerance, Some new results on odd perfect numbers, Pacific J. Math. 57 (1975), 359–364.

[5] Peter Hagis Jr. and Wayne L. McDaniel, A new result concerning the structure of odd perfect numbers, Proc. Amer. Math. Soc. 32 (1972), 13– 15.

[6] H.-J. Kanold, Untersuchungen über ungerade vollkommene Zahlen, J. Reine Angew. Math. 183 (1941), 98–109.

[7] H.-J. Kanold, Sätze über Kreisteilungspolynome und ihre Anwendungen auf einige zahlentheoretische Probleme, I, J. Reine Angew. Math. 187 (1950), 169–182.

[8] Wayne L. McDaniel, The non-existence of odd perfect numbers of a certain form, Arch. Math. (Basel) 21 (1970), 52–53.

[9] Wayne L. McDaniel and P. Hagis Jr., Some results concerning the nonexistence of odd perfect numbers of the form p^{a}M^{2β}, Fibonacci Quart. 13 (1975), 25–28.

[10] T. Nagell, Introduction to Number Theory, Second edition, Chelsea, New York, 1964.

[11] Pace P. Nielsen, An upper bound for odd perfect numbers, Integers 3 (2003), #A14.

[12] R. Steuerwald, Verschärfung einer notwendigen Bedingung für die Existenz einen ungeraden vollkommenen Zahl, S.-B. Bayer. Akad. Wiss. 1937, 69–72.

[13] K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. für Math. 3 (1882), 265–284.

Tomohiro Yamada Department of Mathematics Faculty of Science Kyoto University Kyoto, 606-8502 Japan

e-mail: tyamada@math.kyoto-u.ac.jp