Let <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lfloor x\rfloor $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>⌊</mo> <mi>x</mi> <mo>⌋</mo> </mrow> </math></EquationSource> </InlineEquation> denote the greatest integer less than or equal to a real number x. Given real numbers <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$0<\alpha _1< \alpha _2< \cdots< \alpha _k < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo><</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>α</mi> <mi>k</mi> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> satisfying a certain condition, we show that there are infinitely many positive integers n for which all of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lfloor n^{\alpha _1}\rfloor , \lfloor n^{\alpha _2}\rfloor ,\ldots , \lfloor n^{\alpha _k}\rfloor $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo>⌊</mo> <msup> <mi>n</mi> <msub> <mi>α</mi> <mn>1</mn> </msub> </msup> <mo>⌋</mo> </mrow> <mo>,</mo> <mrow> <mo>⌊</mo> <msup> <mi>n</mi> <msub> <mi>α</mi> <mn>2</mn> </msub> </msup> <mo>⌋</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow> <mo>⌊</mo> <msup> <mi>n</mi> <msub> <mi>α</mi> <mi>k</mi> </msub> </msup> <mo>⌋</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are prime numbers. Our approach relies on establishing a simultaneous equidistribution theorem for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\lfloor n^{\alpha _i}\rfloor $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>⌊</mo> <msup> <mi>n</mi> <msub> <mi>α</mi> <mi>i</mi> </msub> </msup> <mo>⌋</mo> </mrow> </math></EquationSource> </InlineEquation> across k-many arithmetic progressions.

How often are $\lfloor n^{\alpha }\rfloor $ and $\lfloor n^{\beta }\rfloor $ simultaneously primes?

期刊论文
发表日期：

2026年3月31日

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Anup B. Dixit,
Nikhil S. Kumar

摘要

Let $\lfloor x\rfloor $ $⌊ x ⌋$ denote the greatest integer less than or equal to a real number x. Given real numbers $0<\alpha _1< \alpha _2< \cdots< \alpha _k < 1$ $0 < α_{1} < α_{2} < \dots < α_{k} < 1$ satisfying a certain condition, we show that there are infinitely many positive integers n for which all of $\lfloor n^{\alpha _1}\rfloor , \lfloor n^{\alpha _2}\rfloor ,\ldots , \lfloor n^{\alpha _k}\rfloor $ $⌊ n^{α_{1}} ⌋, ⌊ n^{α_{2}} ⌋, \dots, ⌊ n^{α_{k}} ⌋$ are prime numbers. Our approach relies on establishing a simultaneous equidistribution theorem for $\lfloor n^{\alpha _i}\rfloor $ $⌊ n^{α_{i}} ⌋$ across k-many arithmetic progressions.

How often are \(\lfloor n^{\alpha }\rfloor \) and \(\lfloor n^{\beta }\rfloor \) simultaneously primes?

摘要