Let S be a fixed set of primes, and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(X_{l})_{l\ge 1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi>l</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>l</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> be the X-coordinates of the positive integer solutions (X, Y) of the Pell equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$X^2-dY^2 = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>X</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>d</mi> <msup> <mi>Y</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> corresponding to a nonsquare integer <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We show that there are only a finite number of nonsquare integers <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> such that there are at least two different elements of the sequence <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(X_{l})_{l\ge 1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi>l</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>l</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> that can be represented as a sum of S-units with a fixed number of terms. Furthermore, we explicitly solve a particular case in which two of the X-coordinates are products of a power of 2 and a power of 3.

Sums of S-units in X-coordinates of Pell equations

期刊论文
发表日期：

2026年7月24日

查看全文

Parvathi S. Nair,
S. S. Rout

摘要

Let S be a fixed set of primes, and let $(X_{l})_{l\ge 1}$ ${(X_{l})}_{l \geq 1}$ be the X-coordinates of the positive integer solutions (X, Y) of the Pell equation $X^2-dY^2 = 1$ $X^{2} - d Y^{2} = 1$ corresponding to a nonsquare integer $d>1$ $d > 1$ . We show that there are only a finite number of nonsquare integers $d>1$ $d > 1$ such that there are at least two different elements of the sequence $(X_{l})_{l\ge 1}$ ${(X_{l})}_{l \geq 1}$ that can be represented as a sum of S-units with a fixed number of terms. Furthermore, we explicitly solve a particular case in which two of the X-coordinates are products of a power of 2 and a power of 3.