# There are no positive integers such that $a+b^2$, $a^2+b$ are both powers of $5$

Prove that there doesn't exist positive integers $$a$$ and $$b$$ such that $$a^2+b$$ and $$b^2+a$$ are both powers of $$5$$.

Assume on the contrary. Let $$a^2+b=5^x, \quad a+b^2=5^y$$ With $$x,y\ge 1$$. WLOG assume $$x\ge y$$ which means $$a\ge b$$ and $$a+b^2\mid a^2+b\implies a^2+b\equiv 0\pmod{a+b^2}$$

Since $$a\equiv -b^2\pmod {a+b^2}$$ we get $$a^2+b\equiv b^4+b\equiv 0\pmod{a+b^2}$$

Claim $$(a,5)=(b,5)=1$$. Furthermore $$(a,b)=1$$.

Proof Assume $$5\mid a$$ then immediately $$5\mid b$$. Let $$a=5a_1,$$ $$b=5b_1$$ plugging this to the original equation $$5a_1^2+b_1=5^{x-1}$$ Now if $$x=1$$ we get a contradiction. So assume $$x\ge 2$$ clearly $$5\mid b_1$$ meaning $$b_1=5b_2$$. $$a_1^2+b_2=5^{x-2}$$ Again $$x=2$$ is impossible. So $$a_1=5a_2...$$ Cleary this can't go on forever hence $$a_1=a_2=0$$ a contradiction.

If $$p\mid (a,b)$$ then $$p\mid 5$$ meaning $$p=5$$ contradiction.

Back to our result $$a+b^2\mid b(b^3+1)$$ since $$(a+b^2,b)=1$$ we get $$a+b^2\mid b^3+1$$

We're pretty much done. we just need the following claim.

Claim $$5\mid b+1$$.

Proof We know $$5\mid b^3+1$$ Thus $$b^3\equiv -1\pmod 5\implies b^6\equiv 1\pmod 5$$ $$\operatorname{ord}_5b\mid 6$$ but $$\operatorname{ord}_5b \mid 4$$ so $$\operatorname{ord}_5b=2$$ or $$1$$. Surely it's not $$1$$. $$b^2\equiv 1\pmod 5 \implies b\equiv -1 \pmod 5$$ So $$5\mid b+1$$.

All our work till now was in order to use the LTE lemma (Lifting the exponent). Since $$5\nmid b$$ and $$5\mid b+1$$ we get $$\nu_5(a+b^2)\le \nu_5(b^3+1)=\nu_5(b+1)+\nu_5(3)=\nu_5(b+1)$$ In fact since the LHS is a power of $$5$$ we get $$a+b^2\mid b+1 \implies b+b^2\le a+b^2\le b+1$$. Hence $$b\le 0$$ A contradiction.

I just need a verification and if you see a part where I could've used a better method to do it -especially the first claim- please tell me.

What you've done is mostly correct, but your proof of $$\gcd(a,5)=\gcd(b,5)=1$$ has a significant problem. In particular, with your result of $$a_1^2+b_2=5^{x-2}$$, you then wrote

Again $$x=2$$ is impossible. So $$a_1=5a_2...$$

However, you haven't proven, or even indicated, why we require $$a_1=5a_2$$ (e.g., since there's no indication or requirement that $$5 \mid b_2$$). Nonetheless, your claim is correct, with also using the other equation being one way to prove it. Starting as you did to first get

$$5a_1^2 + b_1 = 5^{x-1} \tag{1}\label{eq1A}$$

we can then use that same substitution of $$a = 5a_1$$ and $$b = 5b_1$$ in the second equation to obtain

$$a_1 + 5b_1^2 = 5^{y-1} \tag{2}\label{eq2A}$$

Next, as you did, we have $$x \gt 1$$ and $$y \gt 1$$, so \eqref{eq1A} shows $$b_1 = 5b_2$$ and \eqref{eq2A} shows $$a_1 = 5a_2$$. Using these in both equations and simplifying again gives

$$25a_2^2 + b_2 = 5^{x-2}, \; \; \; a_2 + 25b_2^2 = 5^{y-2} \tag{3}\label{eq3A}$$

This procedure can keep being repeated (actually, we could use that $$25$$ divides $$a_2$$ and $$b_2$$, with higher powers of $$5$$ later, but it's sufficient, as well as simpler & easier, to just use one factor of $$5$$ each time). After doing that $$j$$ times, we end up with

$$5^{j}a_j^2 + b_j = 5^{x-j}, \; \; \; a_j + 5^{j}b_j^2 = 5^{y-j} \tag{4}\label{eq4A}$$

which is clearly impossible for large enough $$j$$ (i.e., since $$x \ge y$$, once $$j = \left\lceil\frac{y}{2}\right\rceil$$).

As for potential improvements in the presentation of your proof, the only ones I suggest are with

$$ord_5b\mid 6$$ but $$ord_5b \mid 4$$ so $$ord_5b=2$$ or $$1$$

First, instead of ord_5b which becomes $$ord_5b$$, so the first part looks like the product of $$3$$ variables, I recommend using \operatorname{ord}_5b instead, with this becoming $$\operatorname{ord}_5b$$, and it also automatically adds some horizontal space before the $$b$$ (with this, to me at least, making it look a bit better). Second, you may wish to explain that the first two parts together means that $$\operatorname{ord}_{5}b \mid (\gcd(6,4) = 2)$$, so that's why $$\operatorname{ord}_{5}b$$ must be either $$2$$ or $$1$$.

Finally, regarding any possibly better methods to use, with your claim that $$5 \mid b + 1$$, rather than using the multiplicative order, instead as Ross Millikan's answer indicates, use that $$5 \nmid a$$ so $$a^2 \equiv 1 \text { or } 4 \pmod{5}$$. Thus, from $$a^2 + b \equiv 0 \pmod{5}$$, we have $$b \equiv 1 \text { or } 4 \pmod{5}$$. If $$b \equiv 1 \pmod{5}$$ then, from $$a + b^2 \equiv 0 \pmod{5}$$, we have $$a \equiv 4 \pmod{5}$$. However, then $$a^2 + b \equiv 1 + 1 \equiv 2 \pmod{5}$$, so $$b \not\equiv 1 \pmod{5}$$. This means $$b \equiv 4 \pmod{5}$$ (and also $$a \equiv 4 \pmod{5})$$.

Second, note that $$b^3 + 1 = (b + 1)(b^2 - b + 1)$$. With $$b \equiv 4 \pmod{5}$$, we have $$b^2 - b + 1 \equiv 1 - 4 + 1 \equiv 3 \pmod{5}$$, so $$\nu_5(b^3 + 1) = \nu_5(b + 1)$$. This is basically what the lifting-the-exponent (LTE) lemma uses itself but, for something as relatively simple as in this case, it may be preferable to more directly prove it instead as I've explained.

Let be by contradiction $$a^2+b=5^x$$ and $$a+b^2=5^y$$ with $$x\lt y$$ (one has $$x\ne y$$ necessarily) so we have $$a^2+b\equiv0\pmod{5^k}$$ for $$k=1,2,\cdots, x$$ and similarly $$a+b^2\equiv0\pmod{5^k}$$.

For $$k=1$$ we have the table of the function $$f(x,y)=x^2+y$$ modulo $$5$$ where one can see that the only $$(a,b)$$ and $$(b,a)$$ giving zero is $$(4,4)$$. Explaining we have $$(x,y)=(1,4),(2,1),(3,1)$$ makes $$0$$ modulo $$5$$ but $$(4,1),(1,2),(1,3)$$ does not.

$$\begin{array}{|c|c|}\hline &1 & 2 & 3 & 4 &0 \\\hline 1 & 2 & 3 &4&0&1\\\hline 2 &0&1&2&3&4 \\\hline 3&0&1&2&3&4\\\hline4&2&3&4&0&1\\\hline0&1&2&3&4&0\\\hline\end{array}$$ It follows that $$a=5A-1$$ and $$b=5B-1$$ (because $$4=-1\pmod5$$). Come back to the beginning we have $$\begin{cases}a^2+b=5^x\\a+b^2=5^y\end{cases}\Rightarrow\begin{cases}5^2A^2-10A+5B=5^x\\5^2B^2-10B+5A=5^y\end{cases}\Rightarrow\begin{cases}5A^2-2A+B=5^{x-1}\\5B^2-2B+A=5^{y-1}\end{cases}$$ from which $$(B-A)\left[5(B+A)-3\right]=5^{y-1}-5^{x-1}=5^{x-1}(5^{y-x}-1)$$ so we have $$B-A=5^{x-1}\\5(B+A)=5^{y-x}+2\hspace1cm(*)$$ The second equation $$(*)$$ leads to the absurd $$2\equiv0\pmod5$$ or $$3\equiv0\pmod5$$. This finishes the proof.

• Good solution. But this method only works because we have $a^2$. The higher the power the harder solving the problem becomes to this method.
– PNT
Commented Dec 1, 2022 at 19:43
• Actually this problem is just a weak version of India TST 2019. You can try solving it using this approach.
– PNT
Commented Dec 1, 2022 at 19:45