The sequence $3, 5, 9, 11, 15, 19, 21, 25, 29, 35,\dots$ consists of odd legsin right triangles with integer side lengths and prime hypotenuse. We show thatthe upper density of this sequence is zero, with logarithmic decay. The sameestimate holds for the sequence of even legs in such triangles. We expect ourupper bound, which involves the Erd\H{o}s--Ford--Tenenbaum constant, to besharp up to a double-logarithmic factor. We also provide a nontrivial lowerbound. Our techniques involve sieve methods, the distribution of Gaussianprimes in narrow sectors, and the Hardy--Ramanujan inequality.
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机译：序列$ 3、5、9、11、15、19、21、25、29、35，\ dots $由直角三角形的奇数腿组成，边长为整数，斜边为素数。我们显示该序列的上密度为零，具有对数衰减。对于这样的三角形中的偶数支脚序列，存在相同的估计。我们期望涉及Erd \ H {o} s-Ford-Tenenbaum常数的上限将锐化为双对数因子。我们还提供了一个重要的下限。我们的技术包括筛分方法，高斯素数在狭窄扇区中的分布以及Hardy-Ramanujan不等式。
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