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So I've been trying to solve this problem for a couple of days now. What I've come up with is this:

By way of contradiction, assume that there are positive integers a and b such that $b^4 + b + 1 = a^4$.

Consider the case when $b \geq a$. Then $a^4 - b^4 = b + 1$, but also $a^4 - b^4 \leq 0$. Thus, $b + 1 \leq 0$. Then, $b \leq -1$, a contradiction.

However, I haven't been able to figure out how to prove it with the case of $b < a$.

Thank you.

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    $\begingroup$ What is $(b+1)^4$? $\endgroup$ – Daniel Fischer Jun 8 '15 at 21:55
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    $\begingroup$ "Without loss of generality": this is wrong. You would only say this if the problem was symmetric with respect to $a$ and $b$. Perhaps what you mean is "Suppose first that $b \ge a$." $\endgroup$ – TonyK Jun 8 '15 at 21:59
  • $\begingroup$ That's an interesting question, I have to admit $\endgroup$ – alkabary Jun 8 '15 at 22:06
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    $\begingroup$ To make @DanielFischer's comment a bit more explicit, the easiest approach to a proof is to observe that the equation requires $b^4$ to be almost equal to $a^4$. How close? Well, with $a$ and $b$ both integers, they can't be any closer than $a-b = 1$. Is that close enough? Does adding $b+1$ to $b^4$ give enough of a "bump" to get you all the way up to $a^4$, if $a = b+1$? $\endgroup$ – Brian Tung Jun 8 '15 at 22:06
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    $\begingroup$ @DayDreamerz Right. Except that "then $a^4 - (b+1)^4$ is positive" is too strong if "positive" means $> 0$. It can, for $a = b+1$, be equal to $0$. But that doesn't influence the argument since $b^4+b+1 - (b+1)^4$ is strictly negative then. If on the other hand "positive" means $\geqslant 0$, then $b = 0,\, a = 1$ make a solution. $\endgroup$ – Daniel Fischer Jun 8 '15 at 22:16
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This boils down to showing that $b^4 + b + 1$ is never the fourth power of an integer.

The difference between consecutive fourth powers is

$$(b+1)^4 - b^4 = 4b^3 + 6b^2 + 4b + 1,$$

which is greater than $b+1$ for all $b > 0$.

Since $b^4 + b + 1$ is a fourth power of an integer, plus $b+1$, you'll never reach another fourth power of an integer by adding $b+1$, because you'll never reach the next fourth power of an integer.

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Indeed, we can show that $b^4+b+1=c^2$ has no solution in positive integers.

This is because if $b>0$ is an integer, then $0<b+1<2b+1\leq 2b^2+1$. Adding $b^4$ to this inequality gives: so $$b^4<c^2=b^4+b+1<b^4+2b^2+1=(b^2+1)^2$$ so $$b^2<c<b^2+1.$$

There are also no solutions with $b\leq -2$, since then:

$$1-2b^2<b+1<0$$ and then add $b^4$ to get: $$(b^2-1)^2<b^4+b+1<b^4$$

There is a $c$ with $b=0,-1$.

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In any case, $b^4-a^4+b+1=0$. Also, $$ (b^4-a^4)=(b^2-a^2)(b^2+a^2)=(b-a)(b+a)(b^2+a^2) $$

So, $$ (b-a)(b+a)(b^2+a^2)= -b-1 $$

So $(b^2+a^2)$ now divides $b+1$. So $b^2+a^2\leq b+1$. The only way this will work is if $a=b=1$. Check that this doesn't satisfy the original equation. So we're done!

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