The 3-Additive Uniqueness of Generalized Pentagonal Numbers for Multiplicative Functions

Poo-Sung Park Department of Mathematics Education Kyungnam University Changwon 517670 South Korea

                     pspark@kyungnam.ac.kr

Abstract The set P of all nonzero generalized pentagonal numbers is k-additively unique for multiplicative functions for fixed k ≥ 2. That is, if a multiplicative function f satisfies the condition

f(a_{1} + a_{2} + ⋯ + a_{k}) = f(a_{1}) + f(a_{2}) + ⋯ + f(a_{k})

for arbitrary a_{1}, a_{2}, …, a_{k} ∈ P, then f is the identity function. The known proof for k = 3 uses the generalized Riemann hypothesis (GRH). Here, we give a proof bypassing GRH.

1 Introduction

In 1992 Spiro [5] coined the term additive uniqueness to describe a curious property of a set E ⊂ ℕ for a set S of arithmetic functions. If a function f ∈ S satisfying the condition

f(x + y) = f(x) + f(y) for all a, b ∈ E

is uniquely determined, she called E an additive uniqueness set for S.

Many mathematicians have been studying and generalizing her seminal work. The concept of k-additive uniqueness is a generalization. A set E is called a k-additive uniqueness set for S, if a function f ∈ S satisfying the condition

f(x_{1} + x_{2} + ⋯ + x_{k}) = f(x_{1}) + f(x_{2}) + ⋯ + f(x_{k}) for all x_{i} ∈ E is uniquely determined.

In 2018, the author and colleagues [2] proved that the set

P = \{P_{m} = \frac{m(3m − 1)}{2} | m ∈ ℤ, m ≠ 0\} = \{1, 2, 5, 7, 12, 15, …\}

of all nonzero generalized pentagonal numbers is an additive uniqueness set for multiplicative functions. Here, a multiplicative function refers to an arithmetic function f : ℕ → ℂ satisfying f(1) = 1 and f(ab) = f(a)f(b) for all a, b with gcd(a, b) = 1. In addition, they [3] extended this result to the set

\{P_{m} = \frac{m(3m − 1)}{2} | m ∈ ℤ, m ≥ 1\} = \{1, 5, 12, 22, 35, 51, …\}

of nonzero ordinary pentagonal numbers.

In 2022, Hasanalizade [1] extended the author’s finding for generalized pentagonal numbers to k-additive uniqueness with k ≥ 3. However, his proof for the case of k = 3 relies on the proposition that every integer > 2 can be written as a sum of three nonzero generalized pentagonal numbers [4, Theorem 4.3], which requires the generalized Riemann hypothesis (GRH). In this article, we present an elementary proof that does not rely on GRH.

2 Main result

Theorem 1. Let f be a multiplicative function and let P be the set of nonzero generalized pentagonal numbers. If

f(a + b + c) = f(a) + f(b) + f(c)

for every a, b, c ∈ P, then f is the identity function.

Proof. We have f(1) = 1 and thus f(3) = 3. Note that

f(5) = f(1) + f(2) + f(2) = 1 + 2 f(2),
f(7) = f(1) + f(1) + f(5) = 3 + 2f f(2).

Then f(2) = 2 by

f(9) = f(2) + f(2) + f(5) = 1 + 4 f(2)
= f(1) + f(1) + f(7) = 5 + 2 f(2).

So, clearly, f(n) = n for n ≤ 9.

We use induction. Assume that f(n) = n for all n < N.

If N = ab with a, b ≥ 2 and gcd(a, b) = 1, then f(N) = f(a) f(b) = ab by the induction hypothesis. So it suffices to consider the case when N is a power of a prime.

Consider the case of N = 2^{r}. We have that 2^{r} = 3m ± 1 for some m according to the parity of r.

If N = 2^{2s+1}, then N = 3m − 1 for some m ∈ ℕ. Note that

P_{m} + P_{−2m+1} + P_{−2m+1}

= m · \frac{3m − 1}{2} + (2m − 1)(3m − 1) + (2m − 1)(3m − 1)

= \frac{3m − 1}{2} · (9m − 4)

and

P_{2m−1} + P_{−2m+1} + P_{−2m+1}
= (2m − 1)(3m − 2) + (2m − 1)(3m − 1) + (2m − 1)(3m − 1)
= (2m − 1)(9m − 4).

Since each factor < 3m − 1 is fixed by f by the induction hypothesis, letting x = f(3m − 1) and y = f(9m − 4), we have that

m · \frac{3m − 1}{2} + 2(2m − 1) x = \frac{3m − 1}{2} y
(2m − 1)(3m − 2) + 2(2m − 1) x = (2m − 1) y.

Thus x = 3m − 1 and y = 9m − 4. So f(2^{2s+1}) = 2^{2s+1}.

If we change m to −m, then we can show that f fixes 9m + 4 and N = 2^{2s} = 3m + 1 in the same way.

Now suppose N = 3^{r}. Then

P_{3^{r−1}} + P_{3^{r−1}} + P_{3^{r−1}} = 3^{r−1} · \frac{3^{r} − 1}{2} + 3^{r−1} · \frac{3^{r} − 1}{2} + 3^{r−1} · \frac{3^{r} − 1}{2} = 3^{r} · \frac{3^{r} − 1}{2},

and thus f(3^{r}) = 3^{r}.

Finally, let N = p^{r} with odd prime p ≠ 3. Assume N = 6m − 1 for some m. Note that

P_{m} + P_{−m} + P_{2m} = \frac{m(3m − 1)}{2} + \frac{m(3m + 1)}{2} + m(6m − 1)
= m(9m − 1)

and

P_{−m} + P_{2m} + P_{2m} = \frac{m(3m + 1)}{2} + m(6m − 1) + m(6m − 1)

= \frac{3m(9m − 1)}{2}.

The first expression leads to an equation

3m + x = y

where x = f(6m − 1) and y = f(9m − 1).

For the second expression, there are two cases according to m. If 9m − 1 is even, then m is odd and

f(\frac{3m(9m − 1)}{2}) = f(3m) f(\frac{9m − 1}{2}) = 3m · \frac{9m − 1}{2},

since 3m and \frac{9m−1}{2} are smaller than N = 6m − 1. If 9m − 1 is odd, then m is even and

f(\frac{3m(9m − 1)}{2}) = f(\frac{3m}{2}) f(9m − 1) = \frac{3m}{2} · f(9m − 1) = \frac{3m}{2} · y.

In both cases, we conclude that x = f(6m − 1) = 6m − 1 and y = f(9m − 1) = 9m − 1.

Similarly, we can show that f(p^{r}) = p^{r} by changing m to −m, when p^{r} = 6m + 1. The proof is complete.

3 Acknowledgment

This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2021R1A2C1092930).

References

[1] E. Hasanalizade, Multiplicative functions k-additive on generalized pentagonal numbers, Integers 22 (2022), #A43.

[2] B. Kim, J. Y. Kim, C. G. Lee, and P.-S. Park, Multiplicative functions additive on generalized pentagonal numbers, C. R. Math. Acad. Sci. Paris 356 (2018), 125–128.

[3] B. Kim, J. Y. Kim, C. G. Lee, and P.-S. Park, Multiplicative functions additive on polygonal numbers, Aequationes Math. 95 (2021), 601–621.

[4] D. Kim, J. Lee and B.-H. Oh, A sum of three nonzero triangular numbers, Int. J. Number Theory 17 (2021), 2279–2300.

[5] C. A. Spiro, Additive uniqueness sets for arithmetic functions, J. Number Theory 42 (1992), 232–246.

2020 Mathematics Subject Classification: Primary 11N64. Keywords: k-additive uniqueness, multiplicative function, generalized pentagonal number.

(Concerned with sequence \underline{A001318}.)

Received April 14 2023; revised version received June 9 2023. Published in Journal of Integer Sequences, June 9 2023.

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