<p>For any <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d \geqslant 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we prove that there exists an integer <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n_0(d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that there exists an <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n \times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> magic square of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d^\text {th}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>d</mi> <mi mathvariant="normal">th</mi> </msup> </math></EquationSource> </InlineEquation> powers for all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n \geqslant n_0(d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <msub> <mi>n</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In particular, we establish the existence of an <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n \times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> magic square of squares for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n \geqslant 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, which settles a conjecture of Várilly-Alvarado. All previous approaches had been based on constructive methods and the existence of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n \times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> magic squares of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$d^\text {th}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>d</mi> <mi mathvariant="normal">th</mi> </msup> </math></EquationSource> </InlineEquation> powers had only been known for sparse values of <i>n</i>. We prove our result by the Hardy-Littlewood circle method, which in this setting essentially reduces the problem to finding a sufficient number of disjoint linearly independent subsets of the columns of the coefficient matrix of the equations defining magic squares. We prove an optimal (up to a constant) lower bound for this quantity.</p>

On the existence of magic squares of powers

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2025年9月23日

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Nick Rome,
Shuntaro Yamagishi

摘要

For any $d \geqslant 2$ $d ⩾ 2$ , we prove that there exists an integer $n_0(d)$ $n_{0} (d)$ such that there exists an $n \times n$ $n \times n$ magic square of $d^\text {th}$ $d^{th}$ powers for all $n \geqslant n_0(d)$ $n ⩾ n_{0} (d)$ . In particular, we establish the existence of an $n \times n$ $n \times n$ magic square of squares for all $n \geqslant 4$ $n ⩾ 4$ , which settles a conjecture of Várilly-Alvarado. All previous approaches had been based on constructive methods and the existence of $n \times n$ $n \times n$ magic squares of $d^\text {th}$ $d^{th}$ powers had only been known for sparse values of n. We prove our result by the Hardy-Littlewood circle method, which in this setting essentially reduces the problem to finding a sufficient number of disjoint linearly independent subsets of the columns of the coefficient matrix of the equations defining magic squares. We prove an optimal (up to a constant) lower bound for this quantity.