We show that every sufficiently large odd integer n can be represented as <Equation ID="Equ18"> <EquationSource Format="TEX">$\begin{aligned} n&= p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^r + p_6^s + p_7^t, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mi>n</mi> </mtd> <mtd columnalign="left"> <mrow> <mo>=</mo> <msubsup> <mi>p</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mn>3</mn> <mn>3</mn> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mn>4</mn> <mn>3</mn> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mn>5</mn> <mi>r</mi> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mn>6</mn> <mi>s</mi> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mn>7</mn> <mi>t</mi> </msubsup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where each <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> is prime, and r, s, t are fixed integers satisfying either <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(r,s,t)=(3,4,t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$t \ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, or <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$3 \le r \le s \le 6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <mi>s</mi> <mo>≤</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$t \ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(r,s)\ne (3,3),(3,4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

On Sums of Two Squares, Two Cubes, and Higher Powers of Primes

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2026年7月24日

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Geovane Matheus Lemes Andrade

摘要

We show that every sufficiently large odd integer n can be represented as $\begin{aligned} n&= p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^r + p_6^s + p_7^t, \end{aligned}$ $\begin{matrix} n & = p_{1}^{2} + p_{2}^{2} + p_{3}^{3} + p_{4}^{3} + p_{5}^{r} + p_{6}^{s} + p_{7}^{t}, \end{matrix}$ where each $p_i$ $p_{i}$ is prime, and r, s, t are fixed integers satisfying either $(r,s,t)=(3,4,t)$ $(r, s, t) = (3, 4, t)$ with $t \ge 5$ $t \geq 5$ , or $3 \le r \le s \le 6$ $3 \leq r \leq s \leq 6$ with $t \ge 4$ $t \geq 4$ and $(r,s)\ne (3,3),(3,4)$ $(r, s) \neq (3, 3), (3, 4)$ .