The sequence 3,5,9,11,15,19,21,25,29,35,… consists of odd legs in right triangles with integer side lengths and prime hypotenuse. We show that the upper density of this sequence is zero, with logarithmic decay. The same estimate holds for the sequence of even legs in such triangles. We expect our upper bound, which involves the ErdH{o}s--Ford--Tenenbaum constant, to be sharp up to a double-logarithmic factor. We also provide a nontrivial lower bound. Our techniques involve sieve methods, the distribution of Gaussian primes 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，…由直角三角形的奇数边组成，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偶数支脚的序列，具有相同的估计。我们希望我们的上限（包括Erd H {o} s-Ford-Tenenbaum常数）可以达到双对数因子。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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