On Numbers Whose Integer Parts of Powers Are Always Composite

Overview

It is known that there are countably many non-integers a > 1 which possess the property that ?a^n? is not a prime for all integers n = 1. There are however very few known explicit examples of such numbers. A Pisot number is an algebraic integer a > 1 all whose conjugates are of modulus less than 1. The degree of a Pisot number is the degree of its minimal polynomial. In this paper we exhibit infinitely many explicit quadratic, cubic, and quartic Pisot numbers a for which ?a^n? is composite for all positive integers n. Moreover, we prove that for every d = 2 where there exist infinitely many Pisot numbers of degree d which have the desired property.

Awards (2)

• Lawrence Technological University: STEM Scholar Award, a tuition scholarship of $19,650 per year, renewable for up to four years and applicable to any major $19,650
• National Security Agency Research Directorate : Third Place Award “Mathematics”

Competition history

• ISEF 2023 Mathematics · Entry MATH040 · Dallas, Texas, United States

Resources

• ISEF abstract page ↗

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