arXiv subjects

Brennan Benfield

Publications and source records attributed to Brennan Benfield.

5 recordsLinked to original sources

arXiv2025

Integers that are not the sum of positive powers

The generalized Waring problem asks exactly which positive integers cannot be expressed as the sum of $j$ positive $k$-th powers? Using computational techniques, this paper refines an approach introduced by Zenkin, establishes results for the individual cases $5 \le k \le 9$, and resolves conjectures of Zenkin and the OEIS. This paper further establishes theoretical results regarding the properties of the sets of integers that are not the sum of $j$ positive $k$-th powers. The notion of Waring's problem is further extended to the finite sets of non-representable numbers where $G(1,k) < j < g(1,k)$. Improved computational techniques and results from Waring's problem are used throughout to catalog the sets of such integers, which are then considered in a general setting.

math.NT↗

arXiv2025

Connecting Zeros in Pisano Periods to Prime Factors of $K$-Fibonacci Numbers

The Fibonacci sequence is periodic modulo every positive integer $m>1$, and perhaps more surprisingly, each period has exactly 1, 2, or 4 zeros that are evenly spaced, which also holds true for more general $K$-Fibonacci sequences. This paper proves several conjectures connecting the zeros in the Pisano period to the prime factors of $K$-Fibonacci numbers. The congruence classes of indices for $K$-Fibonacci numbers that are multiples of the prime factors of $m$ completely determine the number of zeroes in the Pisano period modulo $m$.

math.NT↗

arXiv2024

Fixed points of K-Fibonacci sequences

A $K$-Fibonacci sequence is a binary recurrence sequence where $F_0=0$, $F_1=1$, and $F_n=K\cdot F_{n-1}+F_{n-2}$. These sequences are known to be periodic modulo every positive integer greater than $1$. If the length of one shortest period of a $K$-Fibonacci sequence modulo a positive integer is equal to the modulus, then that positive integer is called a $\textit{fixed point}$. This paper determines the fixed points of $K$-Fibonacci sequences according to the factorization of $K^2+4$ and concludes that if this process is iterated, then every modulus greater than $3$ eventually terminates at a fixed point.

math.NT↗

arXiv2024

End behavior of Ramanujan's taxicab numbers

Generalized taxicab numbers are the smallest positive integers that are the sum of exactly $j$, positive $k$-th powers in exactly $m$ distinct ways. This paper is considers for which values of $m$ does a smallest such integer exist as $j$ gets large. There appear to be only two possible outcomes, leading to curious results like there is no positive integer that can be expressed as the sum of exactly $10$ positive squares in exactly $3$ ways. This paper resolves a number of conjectures found in the OEIS by considering generalized Taxicab numbers in the setting of the theory of partitions.

math.NT↗

arXiv2022

The Fibonacci Sequence is Normal Base 10

In this paper, we show that the concatenation of the Fibonacci sequence is \textit{normal} in base $10$, meaning every string of a given length, $k$, occurs as frequently as every other string of length $k$ (there are as many $1$'s as $2$'s and as many $704$'s and $808$'s). Although we know that almost every number is normal, we can name very few of them. It is still unclear if $e$, $π$, or $\sqrt{2}$ are normal. We show that concatenating the Fibonacci sequence behind a decimal creates a normal number in every base of the form $5^x\times2^y$. We then provide evidence that potentially extends our result to all integer bases, and claim that the Fibonacci concatenation is \textit{absolutely normal}.

math.NT↗