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On the Existence of Solutions of Diophantine Equations Related to Subbalancing Numbers

2025    Sarı, S.; Karadeniz-Gözeri, G.

In this paper, we introduce a new sequence of subbalancing numbers by considering balancing numbers as the values of $D$ in the Diophantine equations provided by subbalancing numbers. For this, first we prove that when the values of the positive integer $D$ are chosen as balancing numbers, there exist integer solutions of the Diophantine equations that correspond to these values of $D$. Then, we obtain the sequence of $B_ m$-subbalancing numbers as the solutions of these Diophantine equations. Moreover, we especially consider $B_3$-subbalancing numbers and obtain various algebraic identities regarding these numbers. Furthermore, we derive some results on the relationship between $B_3$-subbalancing and $b_3$-subbalancing numbers.