<p>In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_819_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="258" /> </InlineMediaObject> <EquationSource Format="TEX">\(f(X)=X^{q(p-1)+1} +\alpha X^{pq}+X^{q+p-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>X</mi> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi>α</mi> <msup> <mi>X</mi> <mrow> <mi mathvariant="italic">pq</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>X</mi> <mrow> <mi>q</mi> <mo>+</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, on finite fields <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_819_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {F}_{q^2}, q=p^k\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mn>2</mn> </msup> </msub> <mo>,</mo> <mi>q</mi> <mo>=</mo> <msup> <mi>p</mi> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, from [A. Rai, R. Gupta, <i>Further results on a class of permutation trinomials</i>, Cryptogr. Commun. 15 (2023), 811–820].</p>