On some Diophantine equations related to square triangular and balancing numbers (Q2889959)

\(T_n=n(n+1)/2\) is called a \textit{triangular number}, where \(n\) is a natural number. It is well known that \(T\) is a triangular number, if and only if \(8T^2+1\) is a perfect square. A few of these numbers are 1, 3, 6, 10, 15, 21, 28, \(\ldots\)NEWLINENEWLINE\(m\) is a \textit{balancing number} if the following equation NEWLINE\[NEWLINE 1+2+\dots+(m-1)=(m+1)+(m+2)+\dots+(m+r) NEWLINE\]NEWLINE is valid for a positive integer \(r\) which is called \textit{balancer}. For example 1, 6, 35 and 204 are balancing numbers with balancers 0, 2, 14 and 84, respectively. Let \(B_n\) denote \(n\)-th balancing number.NEWLINENEWLINESeveral other authors gave a lot of lemmas or theorems belonging to this topic in which the well-known Pell and Lucas numbers appeared certainly.NEWLINENEWLINENEWLINEIn this article the authors investigate the relations between square triangular, special triangular and balancing numbers. For this investigations they deal with positive integer solutions of some Diophantine equations such as NEWLINE\[NEWLINE (x+y\mp 1)^2=8xy, NEWLINE\]NEWLINE \centerline{or} NEWLINE\[NEWLINE (x+y)^2=4x(2y\mp 1),\;(x+y)^2=2x(4y\mp 1),\;(x+y\mp 1)^2=8xy+1, NEWLINE\]NEWLINE \centerline{or} NEWLINE\[NEWLINE x^2+y^2-6xy=\mp 1,\;x^2+y^2-6xy\mp x=0,\;x^2-6xy+y^2\mp 4x-1=0, NEWLINE\]NEWLINE and give several theorems. A few of them are listed below in which triangular or balancing numbers appeared obviously. For more details or more theorems related to triangular, balancing, Pell or Lucas numbers see the original article.NEWLINENEWLINETheorem 1.5. A natural number is a square triangular number, if and only if \(x=B_n^2\) for some natural number \(n\). Let \(y_n\) denote the sequence which is given by NEWLINE\[NEWLINE B_n^2= \frac{y_n(y_n+1)}{2}. NEWLINE\]NEWLINENEWLINENEWLINETheorem 2.3. All positive integer solutions of the Diophantine equation \((x+y-1)^2=8xy\) are given by \((x,y)=(y_n,y_{n+1})\) with \(n\geq 1\).NEWLINENEWLINETheorem 2.7. All positive integer solutions of the equation \((x+y)^2=4x(2y+1)\) are given by \((x,y)=(4B_n^2,4B_nB_{n+1})\) or \((x,y)=(4B_{n+1}^2,4B_nB_{n+1})\) with \(n\geq 1\).