Pythagorean Triangles with Repeated Digits-Repeated Bases

TL;DR

This paper investigates the existence of Pythagorean triangles with legs of the form d^k and hypotenuse with repeated digit bases, extending previous work from base 10 to other number bases, and proves non-existence results.

Contribution

It generalizes prior results by exploring Pythagorean triangles with repeated digit lengths in various bases, providing new theorems on their non-existence.

Findings

No such triangles exist in base 4 with the specified properties.

The paper proves five theorems on non-existence across different bases.

Extends previous base-10 results to other number systems.

Abstract

In 1998, in the winter issue of the journal Mathematics and Computer education (see [1]), Monte Zerger posed the following problem. He had noticed the Pythagorean triple (216,630,666);(216)^2+(630)^2=(666)^2. Note that 216=6^3 and 666 is the hypotenuse length. The question was then, whether there existed a digit d and a positive integer k(other than the above); such that d^k is the leglength of a Pythagorean triangle whose hypotenuse length has exactly k digits, each being equal to d. In 1999, F.Luca and P.Bruckman, answered the above question in the negative. In 2001, K.Zelator(see [2]), took this question further and showed that no Pythagorean triangle exists such that one leg has length d^k, while the other leglegth has exactly k digits in its decimal expansion, with each digit bein equal to d. In this work, we explore the above phenomenon from the point of view of number bases other than 10. We prove five Theorems. As an example, in Theorem 1, we prove that there exists no Pythagorean triangle with one of its leglengths being d^k, while the other leglength having exactly k digits in its base 4 expansion; and with digit being equal to d.