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My attempt: Rearrange to $x^2=2^n + 153$ and with $2^n\geq 2\ $ it follows $x^2 \geq 155\ $. The next square number is 169, so $x = 13$ and $n = 4$. A first solution. Since $2^n$ is even and 153 is odd, $x^2$ will be odd. So any candidate solution will have an even distance of $2m$ from a previous solution and the difference between these solutions is $(x+2m)^2 - x^2 = 4mx + 4m^2$. This difference can be expressed as a difference between two powers of 2, $4mx + 4m^2 = 2^p - 2^n$. My idea was to show that this doesn´t work so that the first solution is the only one.

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  • $\begingroup$ You accidentally wrote 155 a couple of times. $\endgroup$ Sep 16, 2018 at 23:12
  • $\begingroup$ And $z$ for $x$.... Please be more careful in your postings. Otherwise you waste our time and lose credibility. $\endgroup$ Sep 16, 2018 at 23:15
  • $\begingroup$ I apologize for the mistakes and re-edited the question. Sorry. $\endgroup$
    – Parzifal
    Sep 16, 2018 at 23:18
  • $\begingroup$ The Ramanujan-Nagell equation is $x^2-2^n=-7$. Have a look at the literature on that equation, and see whether you can adapt it to your question. You can start with en.wikipedia.org/wiki/Ramanujan–Nagell_equation $\endgroup$ Sep 16, 2018 at 23:20
  • $\begingroup$ You seem to switch from $2^n$ to $n^2$ a lot. Which one do you mean? $\endgroup$
    – lulu
    Sep 16, 2018 at 23:21

2 Answers 2

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$n$ must be even.

$$ 153 = 3^2 \cdot 17 $$

If $$ x^2 - 2 y^2 $$ is divisible by $3,$ then both $x,y$ are divisible by $3.$ Since this $y$ would be a power of $2$ this is impossible.

So $n$ is even, $n=2k,$ and we actually have $$ x^2 - (2^k)^2 = 153 \; , $$ $$ (x+ 2^k) (x-2^k) = 153 $$

Umm. $$ (x+ 2^k) - (x-2^k) = 2^{k+1} $$

This leads to a finite set of possible $x,$ we can factor $153$ as (ordered pairs) $$ 153 \cdot 1 $$ $$ 51 \cdot 3 $$ $$ 17 \cdot 9 $$

$$ 153 - 1 = 152 = 8 \cdot 19 $$

$$ 51 - 3 = 48 = 16 \cdot 3 $$

$$ 17 - 9 = 8 $$

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  • $\begingroup$ Thank you - much appreciated. (It took me some time to figure out why n must be even $\endgroup$
    – Parzifal
    Sep 17, 2018 at 0:22
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Suppose $x \geq 0$. Studying the equation modulo $2$ reveals that $x$ must be odd. Studying it modulo $3$ reveals that $n$ must be even $n=2k$.

$$x^2-2^n=(x-2^k)(x+2^k)=153$$

So $x-2^k$ and $x+2^k$ are two integers that average to $x$ but multiply to $153$. But $153=51(3)=1(153)=17(9)$, so either $x=26$ or $x=77$ or $x=13$. But $x$ is odd, so we must have $x=13$ or $x=77$. But $x=77$ doesn’t work.


Modulo 2: $x^2 \equiv 1$ so $x \equiv 1$.

Modulo $3$: $x^2-(-1)^n \equiv 0$ so $x^2 \equiv (-1)^n$. But since any square is equivalent to $0$ or $1$ modulo $3$. Either $(-1)^n \equiv 0$ (which is obviously impossible) or $(-1)^n \equiv 1$ (which is only possible if $n$ is even).

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  • $\begingroup$ Your explanation of why n is even was very helpful for understanding the solution - thank you. $\endgroup$
    – Parzifal
    Sep 17, 2018 at 0:27
  • $\begingroup$ Glad to help! @Parzifal $\endgroup$ Sep 17, 2018 at 0:30

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