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  1. (, Journal of the London Mathematical Society)
    Abstract In this paper, we investigate‐multimagic squaresof order . These are magic squares that remain magic after raising each element to the th power for all . Given , we consider the problem of establishing the smallest integer for which there existnontrivial‐multimagic squares of order . Previous results on multimagic squares show that for large . We use the Hardy–Littlewood circle method to improve this to The intricate structure of the coefficient matrix poses significant technical challenges for the circle method. We overcome these obstacles by generalizing the class of Diophantine systems amenable to the circle method and demonstrating that the multimagic square system belongs to this class for all . 
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    Free, publicly-accessible full text available September 1, 2026
  2. (, Bulletin of the Australian Mathematical Society)
    Abstract We show that for any setDof at least two digits in a given baseb, almost all even integers taking digits only inDwhen written in basebsatisfy the Goldbach conjecture. More formally, if$$\mathcal {A}$$is the set of numbers whose digits basebare exclusively fromD, almost all elements of$$\mathcal {A}$$satisfy the Goldbach conjecture. Moreover, the number of even integers in$$\mathcal {A}$$which are less thanXand not representable as the sum of two primes is less than$$|\mathcal {A}\cap \{1,\ldots ,X\}|^{1-\delta }$$. 
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    Free, publicly-accessible full text available June 1, 2026
  3. ; ; (, Mathematika)
    Abstract Let and be natural numbers greater or equal to 2. Let be a homogeneous polynomial in variables of degree with integer coefficients , where denotes the inner product, and denotes the Veronese embedding with . Consider a variety in , defined by . In this paper, we examine a set of integer vectors , defined bywhere is a nonsingular form in variables of degree with for some constant depending at most on and . Suppose has a nontrivial integer solution. We confirm that the proportion of integer vectors in , whose associated equation  is everywhere locally soluble, converges to a constant as . Moreover, for each place of , if there exists a nonzero such that and the variety in admits a smooth ‐point, the constant is positive. 
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  4. (, Mathematika)
    Abstract We derive, via the Hardy–Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of nonsingular local solubility. Our polynomials satisfy the condition that . Our conclusions improve on those that would follow from a direct application of the methods of Birch. For example, we show that in many circumstances the expected asymptotic formula holds when and . 
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  5. ; (, Journal of the London Mathematical Society)
    Abstract Let satisfy . Freĭman's theorem shows that when , there exists such that all large integers are represented in the form , with , if and only if diverges. We make this theorem effective by showing that, for each fixed , it suffices to impose the conditionMore is established when the sequence of exponents forms an arithmetic progression. Thus, for example, when and , all large integers are represented in the form , with . 
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  6. ; (, International Mathematics Research Notices)
    Abstract We establish an asymptotic formula for the number of integral solutions of bounded height for pairs of diagonal quartic equations in $26$ or more variables. In certain cases, pairs in $25$ variables can be handled. 
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  7. (, The Quarterly Journal of Mathematics)
    Abstract We obtain bounds on fractional parts of binary forms of the shape $$\Psi(x,y)=\alpha_k x^k+\alpha_l x^ly^{k-l}+\alpha_{l-1}x^{l-1}y^{k-l+1}+\cdots+\alpha_0 y^k$$ with $$\alpha_k,\alpha_l,\ldots,\alpha_0\in{\mathbb R}$$ and $$l\leq k-2.$$ By exploiting recent progress on Vinogradov’s mean value theorem and earlier work on exponential sums over smooth numbers, we derive estimates superior to those obtained hitherto for the best exponent σ, depending on k and $l,$ such that $$ \min_{\substack{0\leq x,y\leq X\\(x,y)\neq (0,0)}}\|\Psi(x,y)\|\leq X^{-\sigma+\epsilon}. $$ 
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  8. (, Mathematical Proceedings of the Cambridge Philosophical Society)
    Abstract When$$k\geqslant 4$$and$$0\leqslant d\leqslant (k-2)/4$$, we consider the system of Diophantine equations\begin{align*}x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\leqslant j\leqslant k,\, j\ne k-d).\end{align*}We show that in this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when$$d=o\!\left(k^{1/4}\right)$$. 
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  9. (, International Mathematics Research Notices)
    Abstract Let $$k_i\ (i=1,2,\ldots ,t)$$ be natural numbers with $$k_1>k_2>\cdots >k_t>0$$, $$k_1\geq 2$$ and $$t<k_1.$$ Given real numbers $$\alpha _{ji}\ (1\leq j\leq t,\ 1\leq i\leq s)$$, we consider polynomials of the shape $$\begin{align*} &\varphi_i(x)=\alpha_{1i}x^{k_1}+\alpha_{2i}x^{k_2}+\cdots+\alpha_{ti}x^{k_t},\end{align*}$$and derive upper bounds for fractional parts of polynomials in the shape $$\begin{align*} &\varphi_1(x_1)+\varphi_2(x_2)+\cdots+\varphi_s(x_s),\end{align*}$$by applying novel mean value estimates related to Vinogradov’s mean value theorem. Our results improve on earlier Theorems of Baker (2017). 
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  10. (, Bulletin of the Australian Mathematical Society)
    Abstract Let$$\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$$be integral linear combinations of elementary symmetric polynomials with$$\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$$, where$$1\le k_1<\cdots . Subject to the condition$$k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$$, we show that there is a paucity of nondiagonal solutions to the Diophantine system$$\varphi _j({\mathbf x})=\varphi _j({\mathbf y})\ (1\le j\le r)$$. 
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