重数重基勾股三角形
Pythagorean Triangles with Repeated Digits-Repeated Bases
摘要：
1998年，在《数学与计算机教育》（Mathematics and Computer Education）杂志的冬季版（见[1]）上，Monte Zerger提出了以下问题。他注意到毕达哥拉斯三元组（216，630，666）：（216）^2+（630）^2=（666）^2。请注意，216=6^3，666是最大长度。接下来的问题是，是否存在一个数字d和一个正整数k（除了上面的数字），使得d^k是一个勾股三角形的边长，该三角形的边长正好有k个数字，每个数字都等于d。1999年，F.Luca和P.Bruckman对上述问题作了否定的回答。在2001年，K.Zelator（见[2]）进一步研究了这个问题，并证明不存在这样的毕达哥拉斯三角形，即一条腿的长度为d^k，而另一条腿的十进制展开式正好有k位数，每一位数都等于d。在这项工作中，我们探讨了上述现象的角度来看，以外的数字基地10。我们证明了五个定理。作为一个例子，在定理1中，我们证明了不存在一个边长为d^k，而另一个边长恰好有k位数的勾股三角形，并且digit等于d。
英文摘要：
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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