Booth Id:
MATH040
Category:
Mathematics
Year:
2023
Finalist Names:
Kim, Aaron (School: Bronx High School of Science)
Abstract:
It is known that there are countably many non-integers α > 1 which possess the property that ⌊α^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 α > 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 α for which ⌊α^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 Won:
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
National Security Agency Research Directorate : Third Place Award “Mathematics”