Every sufficiently large odd integer is a sum of two positive perfect squares and a prime

Abstract

We prove that every sufficiently large odd integer is a sum of two positive squares and a prime. Let R(n) be the number of representations n = x^2 + y^2 + p with x, y >= 1 and p prime. We show that R(n) > 0 for all odd n >= n0 and obtain the asymptotic formula R(n) = S(n) C n / log n, where S(n) > 0 is the singular series and C > 0 is the singular integral. The proof uses the Hardy-Littlewood circle method: on the major arcs we extract the main term, and on the minor arcs we combine an L2 bound for the prime exponential sum (via the Bombieri-Vinogradov theorem, the large sieve, and the Vaughan-Heath-Brown identity) with a fourth moment estimate for the quadratic Weyl sum. A p-adic local analysis, including the 2-adic case, shows S(n) > 0 for odd n. We work with the standard logarithmic weight on primes and pass to the unweighted count by partial summation.