A NOTE ON THE GENERALISED RAMANUJAN–NAGELL EQUATION $x^2=2^m+p^n$

Abstract Let p be a fixed odd prime. We prove the following results for positive integer solutions $(x,m,n)$ of the equation $(*)\ x^2=2^m+p^n$ . (i) If $p \equiv 3 \pmod 8$ , then $(*)$ has only the solution $(p,x,m,n)=(3,5,4,2)$ . (ii) If $p \equiv 5 \pmod 8$ , then $(*)$ has only the solution $(p,x,m,n)=(5,3,2,1)$ . (iii) If $p \equiv 7 \pmod 8$ , then $(*)$ has at most one solution $(x,m,n)$ , except for $p=7$ , $(x,m,n)=(3,1,1)$ and $(9,5,2)$ . Moreover, if $p=2^q-1$ is a Mersenne prime with $p&gt;7$ , where q is an odd prime with $q&gt;3$ , then $(*)$ has exactly one solution $(x,m,n)=(2^q+1,q+2,2)$ . If $p \equiv 7\pmod 8$ , p is not a Mersenne prime and either $p&lt;1.5\times 10^{12}$ or $p&gt;C$ , where C is an effectively computable absolute constant, then <jats:inline-graphic xmlns: