2013 USAJMO Problems/Problem 1

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Problem

Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?

Solution

No, such integers do not exist. This shall be proven by contradiction, by showing that if $a^5b+3$ is a perfect cube then $ab^5+3$ cannot be.

Remark that perfect cubes are always congruent to $0$, $1$, or $-1$ modulo $9$. Therefore, if $a^5b+3\equiv 0,1,\text{ or} -1\pmod{9}$, then $a^5b\equiv 5,6,\text{ or }7\pmod{9}$.

If $a^5b\equiv 6\pmod 9$, then note that $3|b$. (This is because if $3|a$ then $a^5b\equiv 0\pmod 9$.) Therefore $ab^5\equiv 0\pmod 9$ and $ab^5+3\equiv 3\pmod 9$, contradiction.

Otherwise, either $a^5b\equiv 5\pmod 9$ or $a^5b\equiv 7\pmod 9$. Note that since $a^6b^6$ is a perfect sixth power, and since neither $a$ nor $b$ contains a factor of $3$, $a^6b^6\equiv 1\pmod 9$. If $a^5b\equiv 5\pmod 9$, then \[a^6b^6\equiv (a^5b)(ab^5)\equiv 5ab^5\equiv 1\pmod 9\implies ab^5\equiv 2\pmod 9.\] Similarly, if $a^5b\equiv 7\pmod 9$, then \[a^6b^6\equiv (a^5b)(ab^5)\equiv 7ab^5\equiv 1\pmod 9\implies ab^5\equiv 4\pmod 9.\] Therefore $ab^5+3\equiv 5,7\pmod 9$, contradiction.

Therefore no such integers exist.

Solution 2

We shall prove that such integers do not exist via contradiction. Suppose that $a^5b + 3 = x^3$ and $ab^5 + 3 = y^3$ for integers x and y. Rearranging terms gives $a^5b = x^3 - 3$ and $ab^5 = y^3 - 3$. Solving for a and b (by first multiplying the equations together and taking the sixth root) gives a = $(x^3 - 3)^\frac{5}{24} (y^3 - 3)^\frac{-1}{24}$ and b = $(y^3 - 3)^\frac{5}{24} (x^3 - 3)^\frac{-1}{24}$. Consider a prime p in the prime factorization of $x^3 - 3$ and $y^3 - 3$. If it has power $r_1$ in $x^3 - 3$ and power $r_2$ in $y^3 - 3$, then $5r_1$ - $r_2$ is a multiple of 24 and $5r_2$ - $r_1$ also is a multiple of 24.

Adding and subtracting the divisions gives that $r_1$ - $r_2$ divides 12. (actually, $r_1 - r_2$ is a multiple of 4, as you can verify if $\{^{5r_1 - r_2 = 24}_{5r_2 - r_1 = 48}$. So the rest of the proof is invalid.) Because $5r_1$ - $r_2$ also divides 12, $4r_1$ divides 12 and thus $r_1$ divides 3. Repeating this trick for all primes in $x^3 - 3$, we see that $x^3 - 3$ is a perfect cube, say $q^3$. Then $x^3 - q^3 = 3,$ and $(x-q)(x^2 + xq + q^2) = 3$, so that $x - q = 1$ and $x^2 + xq + q^2 = 3$. Clearly, this system of equations has no integer solutions for $x$ or $q$, a contradiction, hence completing the proof.

Therefore no such integers exist.

Solution 3

Let $a^5b+3=x^3$ and $ab^5+3=y^3$. Then, $a^5b=x^3-3$, $ab^5=y^3-3$, and \[(ab)^6=(x^3-3)(y^3-3)\] Now take $\text{mod }9$ (recall that perfect cubes $\equiv -1,0,1\pmod{9}$ and perfect sixth powers $\equiv 0,1\pmod{9}$) on both sides. There are $3\times 3=9$ cases to consider on what values $\text{mod }9$ that $x^3$ and $y^3$ take. Checking these $9$ cases, we see that only $x^3\equiv y^3\equiv 0\pmod{9}$ or $x\equiv y\equiv 0\pmod{3}$ yield a valid residue $\text{mod }9$ (specifically, $(x^3-3)(y^3-3)\equiv 0\pmod{9}$). But this means that $3\mid ab$, so $729\mid (ab)^6$ so \[729\mid (x^3-3)(y^3-3)\iff 729\mid (27x'^3-3)(27y'^3-3)\iff 81\mid (9x'^3-1)(9y'^3-1)\] contradiction.

Solution 4

If $a^5b+3$ is a perfect cube, then $a^5b$ can be one of $5,6,7 \pmod 9$, so $(a^5b)^5 = a^{25} b^5$ can be one of $5^5 \equiv 2$, $6^5 \equiv 0$, or $7^5 \equiv 4 \pmod 9$. If $a$ were divisible by $3$, we'd have $a^5 b \equiv 0 \pmod 9$, which we've ruled out. So $\gcd(a,9) = 1$, which means $a^6 \equiv 1 \pmod 9$, and therefore $a^{25} b^5 \equiv ab^5 \pmod 9$.

We've shown that $a b^5$ can be one of $0, 2, 4 \pmod 9$, so $ab^5 + 3$ can be one of $3, 5, 7 \pmod 9$. None of these are possibilities for a perfect cube, so if $a^5b+3$ is a perfect cube, $ab^5+3$ cannot be.

Solution 5

As in previous solutions, notice $ab^5,a^5b \equiv 5,6,7 \pmod 9$. Now multiplying gives $a^6b^6$, which is only $0,1 \pmod 9$, so after testing all cases we find that $ab^5\equiv a^5b \equiv 6 \mod 9$. Then since $\phi (9) = 6$, $ab^5\equiv \frac{a}{b}\pmod 9$ and $a^5b \equiv \frac{b}{a}\pmod 9$ (Note that $a,b$ cannot be $0\pmod 9$). Thus we find that the inverse of $6$ is itself under modulo $9$, a contradiction.

Solution 6

I claim there are no such a or b such that both expressions are cubes.

Assume to the contrary $a^5b +3$ and $ab^5 + 3$ are cubes.

Lemma 1: If $a^5b +3$ and $ab^5 + 3$ are cubes, then $ab^5, a^5b \equiv 5,7 \pmod 9$

Proof Since cubes are congruent to any of $0, 1, -1 \pmod 9$, $ab^5,a^5b \equiv 5,6,7 \pmod 9$. But if $ab^5 \equiv 6 \pmod 9$, $3|a$, so $a^5b \equiv 0 \pmod 9$, contradiction. A similar argument can be made for $ab^5 \neq 6 \pmod 9$.


Lemma 2: If k is a perfect 6th power, then $k \equiv 0,1 \pmod 9$

Proof: Since cubes are congruent to $0, 1,  -1 \pmod 9$, we can square, and get 6th powers are congruent to $0, 1 \pmod 9$.


Since $ab^5 \cdot a^5b = a^6 b^6 = (ab)^6$, which is a perfect 6th power, by lemma 2, $ab^5 \cdot a^5b \equiv 0,1 \pmod 9$.

But, by lemma 1, $ab^5 \cdot a^5b \equiv 5 \cdot 5, 5 \cdot 7, 7 \cdot 7 \equiv 7, 8, 4 \pmod 9$.

So, $ab^5 \cdot a^5b$, which is an integer, can't go into any of the possible residue classes modulo 9, without breaking one of these lemmas. This is a contradiction, and the proof is complete. $\blacksquare$

-AlexLikeMath

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