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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.14246 |
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| _version_ | 1866914044087107584 |
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| author | Abrego, Ricardo Adonis Caraccioli |
| author_facet | Abrego, Ricardo Adonis Caraccioli |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14246 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Every sufficiently large odd integer is a sum of two positive perfect squares and a prime Abrego, Ricardo Adonis Caraccioli General Mathematics 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. |
| title | Every sufficiently large odd integer is a sum of two positive perfect squares and a prime |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2509.14246 |