Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation> be the set of odd positive integers that can be represented as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p+2^{a^{2}}+2^{b^{2}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>+</mo> <msup> <mn>2</mn> <msup> <mi>a</mi> <mn>2</mn> </msup> </msup> <mo>+</mo> <msup> <mn>2</mn> <msup> <mi>b</mi> <mn>2</mn> </msup> </msup> </mrow> </math></EquationSource> </InlineEquation>, where p is a prime and a, b are positive integers. In 2022, Yuchen Ding proved that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation> has positive lower asymptotic density. In 2024, the authors showed that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n\notin \mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∉</mo> <mi mathvariant="script">V</mi> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n\equiv 293\hspace{-2.84526pt}\pmod {510}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≡</mo> <mn>293</mn> <mspace width="-2.84526pt" /> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>510</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we prove that if <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\{ kn+l : k=0,1,\dots \} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>k</mi> <mi>n</mi> <mo>+</mo> <mi>l</mi> <mo>:</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is an arithmetic progression of odd positive integers with no terms in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$k\ge 510$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>510</mn> </mrow> </math></EquationSource> </InlineEquation> and k has at least four distinct prime factors. Furthermore, these bounds are best possible. This topic goes back to a conjecture of de Polignac from 1849.

On integers not of the form $p+2^{a^{2}} +2^{b^{2}}$

期刊论文
发表日期：

2026年4月25日

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Ji-Zhen Xu,
Yong-Gao Chen

摘要

Let $\mathcal {V}$ $V$ be the set of odd positive integers that can be represented as $p+2^{a^{2}}+2^{b^{2}}$ $p + 2^{a^{2}} + 2^{b^{2}}$ , where p is a prime and a, b are positive integers. In 2022, Yuchen Ding proved that $\mathcal {V}$ $V$ has positive lower asymptotic density. In 2024, the authors showed that $n\notin \mathcal {V}$ $n \notin V$ if $n\equiv 293\hspace{-2.84526pt}\pmod {510}$ $n \equiv 293 (mod 510)$ . In this paper, we prove that if $\{ kn+l : k=0,1,\dots \} $ ${k n + l : k = 0, 1, \dots}$ is an arithmetic progression of odd positive integers with no terms in $\mathcal {V}$ $V$ , then $k\ge 510$ $k \geq 510$ and k has at least four distinct prime factors. Furthermore, these bounds are best possible. This topic goes back to a conjecture of de Polignac from 1849.

On integers not of the form \(p+2^{a^{2}} +2^{b^{2}}\)

摘要