[Paper Review] On the Diophantine equation x^2+2^a.3^b.11^c=y^n

İsmail Naci Cangül, Musa Demırcı+3 more|arXiv (Cornell University)|Jan 3, 2012

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper resolves the Diophantine equation $x^2 + 2^a \cdot 3^b \cdot 11^c = y^n$ for all integers $a, b, c, x, y, n \geq 3$ with $\gcd(x, y) = 1$, using algebraic number theory and primitive divisor theory. It proves that the only solutions occur for $n = 3, 4, 5, 6, 10$, and provides explicit solutions, with the key result being a complete classification of all such solutions under the coprimality condition.

ABSTRACT

In this note, we find all the solutions of the Diophantine equation x^2 +2^a.3^b.11^c=y^n in nonnegative integers a, b, c, x, y, n>= 3 with x and y coprime.

Motivation & Objective

• To classify all nonnegative integer solutions $(x, y, a, b, c, n)$ to the Diophantine equation $x^2 + 2^a \cdot 3^b \cdot 11^c = y^n$ with $n \geq 3$ and $\gcd(x, y) = 1$.
• To extend the known results on exponential Diophantine equations with $S$-unit coefficients, particularly for $S = \{2, 3, 11\}$.
• To determine the full set of solutions when the right-hand side is a perfect $n$-th power and the left-hand side is a sum of a square and an $S$-unit.
• To establish that the only exponents $n \geq 3$ with prime factors in $\{2, 3\}$ for which solutions exist are $n = 3, 4, 6$.

Proposed method

• Transform the equation into cubic and quartic models of elliptic curves for $n = 3$ and $n = 4$, respectively, to compute $\{2, 3, 11\}$-integral points.
• Use the theory of primitive divisors of Lucas sequences to analyze the case $n \geq 5$, leveraging deep number-theoretic properties of linear recurrence sequences.
• Apply modular arithmetic and congruence arguments to eliminate impossible cases, particularly modulo 8, to constrain possible values of exponents and signs.
• Use computational algebra systems (MAGMA and mwrank) to systematically compute and verify integral points on derived elliptic curves.
• Perform case analysis based on the factorization of $y = u^2 + dv^2$ for $d = 1, 2, 3, 6$, depending on the exponent $n$, to reduce the problem to manageable subcases.
• Analyze each case by substituting $v$ as a product of powers of 2, 3, or 11, and reduce to equations over $\{2, 3, 11\}$-integers to find solutions.

Experimental results

Research questions

• RQ1 What are all the solutions to $x^2 + 2^a \cdot 3^b \cdot 11^c = y^n$ in nonnegative integers with $n \geq 3$ and $\gcd(x, y) = 1$?
• RQ2 For which exponents $n \geq 3$ does the equation have solutions when $n$'s prime factors are in $\{2, 3\}$?
• RQ3 How do primitive divisors of Lucas sequences help in resolving exponential Diophantine equations with composite $S$-unit coefficients?
• RQ4 Can the equation be fully solved using a combination of elliptic curve methods and computational algebra systems?

Key findings

• The only solutions for $n = 3$ are listed in Tables 1 and 2, providing a complete classification for this exponent.
• For $n = 4$, the solutions are explicitly given in Table 3, all satisfying the coprimality and exponent conditions.
• For $n = 5$, the solutions are $(x, y, a, b, c) = (1, 3, 1, 0, 2)$ and $(241, 9, 3, 0, 2)$.
• For $n = 6$, the solutions are $(5, 3, 6, 0, 1)$, $(37, 5, 4, 4, 1)$, and $(117, 5, 4, 0, 2)$.
• For $n = 10$, the solution is $(241, 3, 3, 0, 2)$, which also satisfies the $n = 5$ case but is distinct for $n = 10$.
• The only exponents $n \geq 3$ with prime factors in $\{2, 3\}$ for which solutions exist are $n = 3, 4, 6$, as confirmed by the analysis.