Waring-Goldbach problems for one square and higher powers

We prove that every sufficiently large odd integer can be expressed as a sum of one square and fourteen fifth powers, all of primes. In addition, we establish that every sufficiently large even integer can be written as a sum of one square, one biquadrate, and twelve fifth powers of primes.

💡 Research Summary

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The paper addresses a specific instance of the Waring–Goldbach problem: representing large integers as a sum of one square together with higher powers, all variables being prime numbers. The author improves upon the best known result for the case of fifth powers. Previously, it was known (Zhang, Li, Xue, 2024) that every sufficiently large even integer can be expressed as a sum of a prime square and seventeen prime fifth powers. This work reduces the number of required fifth powers to fourteen for odd integers and twelve for even integers, while also allowing a prime biquadrate (fourth power) in the even case.

The proof relies on the Hardy–Littlewood circle method. The author defines exponential sums over primes, (f_k(\alpha)=\sum_{p\le P_k} (\log p),e(\alpha p^k)) with (P_k=n^{1/k}), and introduces auxiliary sums (g_j(\alpha)) built from carefully chosen exponents (\lambda_j). These parameters are taken from earlier work of Kawada and Wooley and are tuned so that the product (G(\alpha)=g_9(\alpha)^2\prod_{j=1}^{8} g_j(\alpha)) captures the contribution of fourteen (or twelve) fifth‑power variables.

Two main generating functions are considered: (F_1(\alpha)=f_2(\alpha)f_5(\alpha)^4 G(\alpha)) for the odd case, and (F_2(\alpha)=f_2(\alpha)f_4(\alpha)f_5(\alpha)^2 G(\alpha)) for the even case. The number of representations of a given integer (n) is expressed as the Fourier coefficient (\nu_j(n)=\int_0^1 F_j(\alpha) e(-\alpha n),d\alpha).

The integration domain is split into major arcs (\mathfrak M) (rational approximations with denominator (q\le L B), where (L=\log n)) and minor arcs (\mathfrak m). On the major arcs, Lemma 2 provides a precise approximation of each (f_k(\alpha)) by a product of a Kloosterman-type sum (S_k(q,a)) and a smooth function (v_{k,\eta}(\beta)). This yields (F_j(\alpha)=\varphi(q)^{-\ell_j} U_j(q,a) w_j(\beta)+O(n^{1+\Theta_j}L^{-4B})), with (\ell_1=15), (\ell_2=14). After integrating over (\beta) and summing over (q), the main term becomes a convergent arithmetic series (S_j(n)=\sum_{q\ge1} A_{n,j}(q)\varphi(q)^{-\ell_j}), where (A_{n,j}(q)) is essentially a normalized sum of the (U_j(q,a)). Using the Cauchy–Davenport–Vaughan theorem, the author shows that each Euler factor of (S_j(n)) is positive, guaranteeing (S_j(n)\gg1). Consequently, (\int_{\mathfrak M}F_j(\alpha) e(-\alpha n),d\alpha \gg n^{\Theta_j}), with (\Theta_1=3/10+\Lambda/5) and (\Theta_2=3/20+\Lambda/5), where (\Lambda) is a small constant derived from the (\lambda_j).

The treatment of the minor arcs is more delicate. Lemma 4 establishes an (L^2) bound for products of three exponential sums (f_{k_i}) when the exponents satisfy (\sum 1/k_i\ge 3/5). This yields (\int_0^1 |f_2 f_{k_1} f_{k_2} f_{k_3}|^2 d\alpha \ll n^{2(1/k_1+1/k_2+1/k_3)+\varepsilon}). Applying Hölder’s inequality, the author controls the contribution of regions where (|f_5(\alpha)|) is unusually small, defining sets (E_1) and (E_2). The measure of these sets is negligible, and the corresponding integrals are bounded by (n^{\Theta_j-\tau}) for a large (\tau).

For the remaining part of the minor arcs, the author invokes a recent Vinogradov mean‑value theorem (Kumchev–Wooley, 2017) together with Kumchev’s bounds for Weyl sums over primes (2006). These results give a pointwise estimate (|f_k(\alpha)| \ll P_k L^C \Upsilon(\alpha)^{1/2-\varepsilon}), where (\Upsilon(\alpha)=(q+n|q\alpha-a|)^{-1}) for (\alpha) lying in a small neighbourhood of a rational (a/q). This leads to the bound (|F_j(\alpha)| \ll n^{1+\Theta_j} L^{5C} \Upsilon(\alpha)^{39/16}). Integrating (\Upsilon(\alpha)^{39/16}) over the complement of the major arcs yields a contribution of order (n^{\Theta_j} L^{-1}) provided the parameter (B) is chosen sufficiently large (Lemma 6). Hence the total minor‑arc contribution satisfies (\int_{\mathfrak m} F_j(\alpha) e(-\alpha n) d\alpha \ll n^{\Theta_j} L^{-1}).

Putting together the lower bound from the major arcs and the upper bound from the minor arcs, the author obtains (\nu_j(n) \gg n^{\Theta_j}) for all sufficiently large (n) of the appropriate parity. Since (\nu_j(n)) counts weighted prime solutions of the original Diophantine equations, a positive lower bound guarantees the existence of at least one representation with all variables prime. This completes the proof of Theorem 1.

In summary, the paper achieves two significant advances:

1. It reduces the number of required prime fifth powers from 17 to 14 (odd case) and 12 (even case) while adding a prime biquadrate in the even case.
2. It introduces a refined combination of major‑arc analysis, modern Vinogradov mean‑value estimates, and a pruning argument that may be adaptable to other exponents (k) in the Waring–Goldbach framework.

The work is supported by CAPES (Brazil) and acknowledges the University of Brasília’s graduate program. The references list foundational contributions ranging from Stanley’s early 1930s work to recent advances by Kumchev, Wooley, and others, illustrating the deep historical context of the problem.