Beschreibung:
<p>We improve a criterion of Inkeri and show that if there is a solution to Catalan’s equation <disp-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript p Baseline minus y Superscript q Baseline equals plus-or-minus 1 comma">
<mml:semantics>
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mi>p</mml:mi>
</mml:msup>
<mml:mo>−<!-- − --></mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>q</mml:mi>
</mml:msup>
<mml:mo>=</mml:mo>
<mml:mo>±<!-- ± --></mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>,</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\begin{equation}x^p-y^q=\pm 1,\end{equation}</mml:annotation>
</mml:semantics>
</mml:math>
</disp-formula>
with <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p">
<mml:semantics>
<mml:mi>p</mml:mi>
<mml:annotation encoding="application/x-tex">p</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> and <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q">
<mml:semantics>
<mml:mi>q</mml:mi>
<mml:annotation encoding="application/x-tex">q</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> prime numbers greater than 3 and both congruent to 3 <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal m normal o normal d 4 right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="normal">m</mml:mi>
<mml:mi mathvariant="normal">o</mml:mi>
<mml:mi mathvariant="normal">d</mml:mi>
</mml:mrow>
<mml:mspace width="thinmathspace" />
<mml:mn>4</mml:mn>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">(\mathrm {mod}\,4)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, then <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p">
<mml:semantics>
<mml:mi>p</mml:mi>
<mml:annotation encoding="application/x-tex">p</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> and <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q">
<mml:semantics>
<mml:mi>q</mml:mi>
<mml:annotation encoding="application/x-tex">q</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> form a double Wieferich pair. Further, we refine a result of Schwarz to obtain similar criteria when only one of the exponents is congruent to 3 <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal m normal o normal d 4 right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="normal">m</mml:mi>
<mml:mi mathvariant="normal">o</mml:mi>
<mml:mi mathvariant="normal">d</mml:mi>
</mml:mrow>
<mml:mspace width="thinmathspace" />
<mml:mn>4</mml:mn>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">(\mathrm {mod}\,4)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. Indeed, in light of the results proved here it is reasonable to suppose that if <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q identical-to 3">
<mml:semantics>
<mml:mrow>
<mml:mi>q</mml:mi>
<mml:mo>≡<!-- ≡ --></mml:mo>
<mml:mn>3</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">q\equiv 3</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal m normal o normal d 4 right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="normal">m</mml:mi>
<mml:mi mathvariant="normal">o</mml:mi>
<mml:mi mathvariant="normal">d</mml:mi>
</mml:mrow>
<mml:mspace width="thinmathspace" />
<mml:mn>4</mml:mn>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">(\mathrm {mod}\,4)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, then <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p">
<mml:semantics>
<mml:mi>p</mml:mi>
<mml:annotation encoding="application/x-tex">p</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> and <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q">
<mml:semantics>
<mml:mi>q</mml:mi>
<mml:annotation encoding="application/x-tex">q</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> form a double Wieferich pair.</p>