[Paper Review] On practical numbers of some special forms

Li-Yuan Wang, Zhi‐Wei Sun|arXiv (Cornell University)|Sep 3, 2018

Analytic Number Theory Research5 references3 citations

TL;DR

This paper establishes that for certain quadratic forms $n^2 + bn + c$ with $b \geq 0$, $c > 0$, if the form yields a practical number for some $n > 1$, then it produces infinitely many practical numbers. It further proves the existence of infinitely many practical numbers of the form $q^4 + 2$ with $q$ practical, and constructs infinitely many practical Pythagorean triples with GCD 4 or 6, using cyclotomic polynomial identities and inductive arguments on multiplicative structure.

ABSTRACT

In this paper we study practical numbers of some special forms. For any integers $b\ge0$ and $c>0$, we show that if $n^2+bn+c$ is practical for some integer $n>1$, then there are infinitely many nonnegative integers $n$ with $n^2+bn+c$ practical. We also prove that there are infinitely many practical numbers of the form $q^4+2$ with $q$ practical, and that there are infinitely many practical Pythagorean triples $(a,b,c)$ with $\gcd(a,b,c)=6$ (or $\gcd(a,b,c)=4$).

Motivation & Objective

• To investigate the infinitude of practical numbers in quadratic forms $n^2 + bn + c$ under specific arithmetic conditions on $b$ and $c$.
• To establish the existence of infinitely many practical numbers of the form $q^4 + 2$ where $q$ is also practical.
• To construct and prove the existence of infinitely many practical Pythagorean triples with GCD 4 or 6.
• To extend Melfi’s cyclotomic method to prove practicality of specific exponential forms like $2^{35 \cdot 3^k + 1} + 2$.

Proposed method

• Uses the structure theorem for practical numbers: a number is practical iff its prime factors satisfy $p_j - 1 \leq \sigma(p_1^{a_1} \cdots p_{j-1}^{a_{j-1}})$ with $p_1 = 2$.
• Applies Melfi’s cyclotomic method, leveraging identities involving $\Phi_d(x)$ to factorize and bound the size of cyclotomic polynomials.
• Employs induction on $k$ to prove that $m_k = 2^{35 \cdot 3^k + 1} + 2$ is practical for all $k \geq 0$, using bounds on $\Phi_6(x), \Phi_{30}(x), \Phi_{42}(x), \Phi_{210}(x)$.
• Uses the identity $x^n - 1 = \prod_{d|n} \Phi_d(x)$ to decompose $x^{210} - 1$ and relate $m_{k+1}$ to $m_k$ via cyclotomic factors.
• Applies Lemma 2.1 (product of practical numbers with bounded multiplicative growth remains practical) to propagate practicality through recursive factorizations.
• Constructs explicit families of Pythagorean triples via $a_k = 2(3^{3^k \cdot 70} - 1)$, $b_k = 4 \cdot 3^{3^k \cdot 35}$, $c_k = 2(3^{3^k \cdot 70} + 1)$, and proves their practicality inductively.

Experimental results

Research questions

• RQ1 Under what conditions on $b$ and $c$ does the quadratic form $n^2 + bn + c$ produce infinitely many practical numbers?
• RQ2 Are there infinitely many practical numbers of the form $q^4 + 2$ where $q$ is also practical?
• RQ3 Can infinitely many practical Pythagorean triples be constructed with $\gcd(a,b,c) = 4$ or $6$?
• RQ4 Does the form $2^{35 \cdot 3^k + 1} + 2$ yield practical numbers for all $k \geq 0$?
• RQ5 Can cyclotomic polynomial factorizations be used to prove practicality of exponential forms via inductive arguments?

Key findings

• If $n^2 + bn + c$ is practical for some $n > 1$, then there are infinitely many $n \in \mathbb{N}$ such that $n^2 + bn + c$ is practical.
• For $b = 20$, the set $S_{20} = \{1 \leq c \leq 100 : c \not\equiv 2,5 \pmod{12}\}$ ensures that $n^2 + 20n + c$ is practical for infinitely many $n$.
• The number $2^{35 \cdot 3^k + 1} + 2$ is practical for every $k \geq 0$, proving the existence of infinitely many practical $q$ such that $q^4 + 2$ is also practical.
• There are infinitely many practical Pythagorean triples with $\gcd(a,b,c) = 4$, constructed via $a_k = 2(3^{3^k \cdot 70} - 1)$, $b_k = 4 \cdot 3^{3^k \cdot 35}$, $c_k = 2(3^{3^k \cdot 70} + 1)$.
• There are infinitely many practical Pythagorean triples with $\gcd(a,b,c) = 6$, constructed via $x_k = 3(3^{3^k \cdot 70} - 1)$, $y_k = 6 \cdot 3^{3^k \cdot 35}$, $z_k = 3(3^{3^k \cdot 70} + 1)$, with all terms proven practical by induction.
• The cyclotomic factorization $x^{210} - 1 = (x^{105} - 1) \cdot \Phi_6(x) \Phi_{30}(x) \Phi_{42}(x) \Phi_{210}(x)$ is used to bound and compare sizes of factors, enabling inductive proof of practicality.