On the Diophantine equation \(x^2 - 4p^m = \pm y^n\) (Q442163)

In the present paper under review, the authors consider the Diophantine equations NEWLINE\[NEWLINEx^2 - 4p^m = \pm y^n,NEWLINE\]NEWLINE where \(x, y, m, n\) are positive integers. They prove that if \(p\geq 11\) is a prime, \(n, m\geq 3\) are odd integers such that NEWLINE\[NEWLINE n\leq 4\times 10^6 p^{3/2}(\log 4p)^2 (R_p + 0.6), NEWLINE\]NEWLINE where \(R_p\) is the regulator of the quadratic field \(\mathbb{Q}(\sqrt{p})\), then the Diophantine equations NEWLINE\[NEWLINEx^2 - 4p^m = \pm y^nNEWLINE\]NEWLINE have solutions. For the proof, they use properties of the quadratic field \(\mathbb{Q}(\sqrt{p})\) and Baker's method. NEWLINENEWLINEUnfortunately, the paper is not well-written. For example, Lemma 4 is not well quoted. Moreover, reference [7] is published in Acta Arithmetica, not in the Journal of Number Theory.