We show that if c is a positive integer satisfying 
$$2c-1=3p^l\ \hbox{or}\ 2c-1=5p^l$$

 with p prime and l positive integer, then the equation

$${x^2 + (2c-1)^m}=c^n$$

 has only the positive integer solution 
$$(x,m,n)=(c-1,1,2)$$


without any congruence condition on a prime p.

On The Generalized Ramanujan–Nagell Equation $$x^2+(2c-1)^m=c^n$$x2+(2c-1)m=cn

Let a, b, c be pairwise relatively prime positive integers such that $$a^2 + b^4=c^2$$
 and b is odd. Then we show that the equation of the title has only one positive integer solution $$(x, m, n)=(a, 4, 2)$$
 under some conditions.

On the Diophantine equation $$\varvec{x^2+b^m=c^n}$$ x 2 + b m = c n with $$\varvec{a^2+b^4=c^2}$$ a 2 + b 4 = c 2

We make use of elementary theory of number and Fermat method of infinite descent so as toobtain somenecessary conditions if the Diophantine equations x 4±my 8=z 2 and x 8±my 8=z 2 have positive Integer solutions that fit (x,y)=1 m.

On the Diophantine Equation

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On the Diophantine Equation