Pythagorean Triangles with Repeated Digits-Repeated Bases.
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.
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