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On some properties of the Diophantine equations a 2 = b 2 ± bc + c 2

4a2 = (2c + b)2 + 3b2. (2) Throughout this paper, a solution of the Diophantine equation (1) or (2) would be represented by the ordered triplet (a, b, c) with c > b. Thus, for example, (7, 3, 8) is a solution of equation (1), and (7, 3, 5) is a solution of equation (2). Note that, for either of the Diophantine equations (1) and (2), if (a, b, c) is a solution, so also is (ka, kb, kc) for any (positive) constant k. Two solutions would be called independent if and only if one is not a constant multiple of the other. For example, (7, 3, 8) and (7, 5, 8) are two independent solutions of the equation (1).

The concept of the Smarandache function related triangles was introduced by Sastry (2000). Denoting by T(a, b, c) the triangle ∆ABC with sides BC = a, CA = b, AB = c, we have the following definition.
Later, the idea was extended to the case of the pseudo Smarandache function, Z(n), by Ashbacher (2000), where the function Z(n), due to Kashihara (1996), is defined as follows : Of particular interest is the study of the 60 degrees and 120 degrees Smarandache function related and pseudo Smarandache function related triangles. It has been shown by Majumdar (2010) that the triangle ∆ABC with ∠CAB = 60 o satisfies the Diophantine equation and the triangle with ∠CAB = 120 o satisfies the Diophantine equation Throughout this paper, a solution of the Diophantine equation (1) or (2) would be represented by the ordered triplet (a, b, c) with c > b. Thus, for example, (7, 3, 8) is a solution of equation (1), and (7, 3, 5) is a solution of equation (2). Note that, for either of the Diophantine equations (1) and (2) From (1) and (2), we see that they are particular cases of the Diophantine equation However, as has been pointed out by Majumdar (2018), neither (1) nor (2) is equivalent to the equation (3). For example, corresponding to x = 14, there are three solutions of the Diophantine equation (3), namely, (14, 13, 3), (14,11,5) and (14, 2, 8), whereas corresponding to a = 7, the Diophantine equation (1) has two solutions and (2) has only one solution. For more on equation (3), the reader is referred to the recent paper of Majumdar (2020).
Note that, the Diophantine equations (1) and (2) are equivalent to the following two Diophantine equations respectively In connection with the Diophantine equation (4), we have the following five results.
Proof: A cumbersome calculation shows that The above identity proves the lemma. Proof: Tedious calculations give The identity above establishes the lemma.
Lemma 5: Let (a, b, c) be a solution of the Diophantine equation (4) with c < 0. Then, (a, b, bc) is also a solution of (4).
The following four lemmas deal with the Diophantine equation (5). Lemma 7: Let (a, b, c) be a solution of the Diophantine equation (5) is also its solution.
Proof: It is straightforward to deduce that which, in turn, proves the lemma. Proof: To prove the lemma, we need to show that A time-consuming simplification would verify the above identity.
Lemma 9: Let (a, b, c) be a solution of the Diophantine equation (5) with b < 0. Then, (a, -b, c + b) is also a solution of (4).