We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes \(p-1\) as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.

中文翻译：

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几乎素数平方的加性问题

我们证明，每个足够大的整数都是一个素数和两个几乎素数平方的和，也是一个光滑数和两个几乎素数平方的和。这种表示的数量是预期的数量级。我们同样将移位素数\(p-1\)的表示视为两个几乎素数平方的和。这些方法涉及解析、自守和代数参数的组合，以高度均匀地处理受限二元二次形式的表示。