Difference between revisions of "2022 IMO Problems/Problem 5"

Revision as of 21:40, 18 June 2025

Problem

Find all triples $(a,b,p)$ of positive integers with $p$ prime and $a^p = b! + p$

Video solution

https://youtu.be/d09PtqRSOuA

https://www.youtube.com/watch?v=-AII0ldyDww [Video contains solutions to all day 2 problems]

Solution

Case 1: $b < p$

Since $b!$ is indivisible by $p$ , then $a$ must also be indivisible by $p$ .

If $a \le b$ , then $a^p-b!$ is divisible by $a$ , so $a$ must be a divisor of $p$ , but $a=1$ obviously has no solutions and we ruled out $a=p$ already. For $a > b$ , let's show that there are no solutions using simple inequalities.

If $b < a < p$ , then $b! \le (a-1)! \le (a-1)^{a-1}$ and $a^p = (a-1+1)^p \ge (a-1)^p + p (a-1)^{p-1}$ by throwing away the remaining (non-negative) terms of binomial theorem. For any solution, $(a-1)^p + p (a-1)^{p-1} \le a^p = b!+p \le (a-1)^{a-1} + p$ , which is impossible for $a-1 > 1$ . That leaves us with $a=2$ and $b=1$ , but $2^n > n+1$ for any integer $n \ge 2$ (proof by induction), so there are no solutions.

If $b < p < a$ , RHS is at most $(p-1)^{p-1}+p$ and LHS is at least $(p+1)^p \ge p^p+p$ (again from binomial theorem), which gives no solutions as well.

Case 2: $p \le b < 2p$

Since $b!$ is divisible by $p$ , then $a$ must also be divisible by $p$ .

In addition, RHS is at most $(2p-1)!+p = p \prod_{i=1}^{p-1} (p-i)(p+i) + p < p^{2p-1} + p \le p^{2p}$ , so $a < p^2$ . We may write $a=pc$ , where $1 \le c < p$ .

Since $c$ is a divisor of $a^p$ and $b!$ it must also be a divisor of $p$ , so $c=1$ and $a=p$ . We're looking for solutions of $p^{p-1}-1 = b!/p$ .

Let's factorise $p^{p-1}-1$ : if $p-1 = 2^k \cdot n$ with $n$ odd and $k > 0$ , it's

$(p^n-1) \cdot (p^n+1) \cdot (p^{2n}+1) \cdot \ldots \cdot (p^{2^{k-1} n}+1) = (p-1) \cdot (1+p+\ldots+p^{n-1}) \cdot (p+1) \cdot (1-p+\ldots-p^{n-2}+p^{n-1}) \cdot (p^{2n}+1) \cdot \ldots \cdot (p^{2^{k-1} n}+1) \,.$

Since $(1+p+\ldots+p^{n-1})$ and $(1-p+\ldots-p^{n-2}+p^{n-1})$ contain an odd number of odd terms (remember the assumption $k > 0$ aka $p \neq 2$ ), they're odd. Also, $p^2 \equiv 1$ modulo $4$ , so $(p^{2n}+1)$ and each following term is even but indivisible by $4$ . The highest power of $2$ dividing $p^{p-1}-1$ is therefore $2^{k + l + k-1}$ where $2^l$ is the highest power dividing $p+1$ .

In comparison, $(p+1)!/p$ has factors $(p+1)$ , $(2^k n)$ $(2^{k-1} n)$ etc (up to $2n$ ), and $(p-1)/2-k$ other even factors, so it's divisible at least by $2^{l+k(k+1)/2+(p-1)/2-k}$ . Since $l+k(k+1)/2+(p-1)/2-k > l+2k-1$ for $p \ge 7$ , the only possible solutions have $p < 7$ or $b = p$ .

If $b=p$ , we reuse the inequalities $p^{p-1} \ge (p-1)^{p-1}+p-1$ and $b!/p \le (p-1)^{p-1}$ to show that there are no solutions for $p > 2$ .

Finally, $5^5-5 = 3120$ isn't a factorial, $3^3-3 = 24 = 4!$ and $2^2-2 = 2 = 2!$ .

Case 3: $2p \le b$

Just like in case 2, $b!$ is divisible by $p$ so $a$ must also be divisible by $p$ . However, $b!$ and $a^p$ are also both divisible by $p^2$ , so remainders modulo $p^2$ tell us that no solutions exist.

Conclusion:

The only solutions are $(a,b,p) = (2,2,2), (3,4,3)$ .

Solution 2

I considered the cases:

1) If $a$ is even, then it must:

  1.a) \(b \neq 1\) and \(p > 2\)
  1.b) \(b!\) is even and \(p = 2\)

2) If $a$ is odd, then it must:

  2.a) \(b \neq 1\) and \(p = 2\)
  2.b) \(b!\) is even and \(p > 2\)

Examining 1.a), we end up with an equation of the form $a^p = p+1$, which has no integer solutions.

In case 1.b), $b!$ can take values: 2, 6, 24, 120, 720, ..., so $b!+2$ takes values: 4, 8, 26, 122, 722, .... We observe that the only perfect square is 4 among the possible cases, as for $b \geq 5$, the result ends in 2, which is not a perfect square. Therefore, we have the triple $(2,2,2)$.

Case 2.a) yields $a^2 = 3$, which is rejected.

Examining the last case 2.b), we have for $b!$ the values: 2, 6, 24, 120, 720, 5040,

  i) \(b! = 2\), then \(a^p = 2+p\), which is rejected.
  ii) \(b! = 6\), then \(a^p = 6+p\), also does not give integer solutions.
  iii) \(b! = 24\), then \(a^p = 24+p\), giving a solution for \(p=3\). Thus, we have the triple \((3,4,3)\).
  iv) \(b \geq 5\):
     \(a^3 = b! + 3\), so \(a\) must end in 3 or 7, which does not give solutions.
     \(a^5 = b! + 5\), so \(a\) must end in 5, also not giving solutions.
     \(a^7 = b! + 7\), so \(a\) must end in 3, again not giving solutions.
     \(a^{11} = b! + 11\), no \(a \in \mathbb{Z^+}\) satisfies this.
     \(a^{19} = b! + 19\), similarly, no solutions.
     Other prime exponents reduce to these, as their last digit is one of \(\{1,3,7,9\}\)

The analysis of the last case is incomplete, which is why I wasn't initially sure about the number of triples. Therefore, with this approach (which is not strictly documented), we find the triples: $(2,2,2), (3,4,3)$.

Solution 3(Unfinished)

Consider $b \geq a$ . Then, $b!$ must have a factor of $a$ . Since $a\mid a^{p}$ and $a\mid b!$ , $a\mid p$ . But $p$ is prime so $a$ can only be $1$ or the set of prime numbers. If $a = 1$ , then $b! + p = 1$ which is impossible since $b$ is a positive integer and so is $p$ . Therefore, $a$ must be the set of prime numbers specifically $a = p$ . This means $a^{a} = b! + a$ . We can rearrange this to solve this Diophantine Equation: $a^{a} - a = b!$ with $a$ being a prime number. Wilson's Theorem states that if $p$ is a prime number, then $(p - 1)! \equiv -1(\mod p)$ . This motivates us to consider different cases.

Case 1: $b = k - 1$ where $k$ is a prime

This means $a^{a} - a \equiv -1(\mod k) = kq - 1 \implies a\mid kq - 1 \implies kq \equiv 1(\mod a) = ta + 1$ . Note that $a^{a} - a$ is always even. Thus, $kq$ is odd. Let $kq = 2c + 1$ . Here, we clearly see that $2c = ta$ . Here, we only consider $a = 2, c = t$ . Note that $a = 2$ does indeed lead a solution of $(a, b, p) = (2, 2, 2)$ so we have found our first solution triple.

Now, we go back to our original equation: $a^{a} - a = b!, b = k - 1$ , $k$ is a prime. Note that $k \geq 5$ . Consider $k = 5$ . Then, $a^{a} - a = 24 \implies a\mid 24$ . It is immediately obvious that $a = 3$ is the only solution which yields a solution of $(a, b, p) = (3, 4, 3)$ . We will prove that is the second and last solution triple.

Case 1.1: $a > k$

Then, $a^{a} - a > k^{k} - k \implies kq - 1 > k^{k} - k \implies kq + k - 1 > k^{k}$ . Since $k$ is a prime, we have $q + 1 - \frac{1}{k} > k^{k - 1}$ . Note that if $q < k$ , then this inequality will obviously not hold as all terms are strictly less than $k$ . If $q \geq k$ , then $k^{2} + k - 1 > k^{k}$ . Using the same trick and dividing by $k^{2}$ , we get a similar contradiction. Therefore, if $a > k$ , we have no solution.

Case 1.2: $a < k$

Note that $a = k$ will lead a contradiction in the modular arithmetic. Note that $k \geq 7$ . We have $a^{a} - a \leq 7q - 1$ . Notice $a^{a} - a < k^{k} - k$ . It is immediately obvious that only $7q - 1 \leq k^{k} - k$ will hold solutions(If $7q - 1 > k^{k} - k$ , we can use a similar trick as above to prove that nothing will work). But note that $k^{k} - k \geq 7^{7} - 7$ which means $7q - 1 \leq 7^{7} - 7$ . It is now obvious that $q \leq 7^{6} - \frac{6}{7}$ \implies $q \leq 7^{5}$ . Recall that $b \geq a$ . Thus, $k - 1 \geq a$ .(I have not finished proving why this doesn't work.)

Case 1.3: $b$ is anything but $k - 1$ where $k$ is a prime

Recall that we have $a^{a} - a = b!$ . Notice that $b = k$ will lead to a contradiction because $a \geq 3 \implies a^{a} - a$ is composite. Therefore, $b = k + \alpha$ where $\alpha$ is a constant of anything but $-1, 0$ . Let $r$ be a prime that divides $(k + \alpha)!$ . Thus, $r \mid a^{a} - a = a(a^{a - 1} - 1)$ . Therefore, we know that $r \mid k + \alpha - i$ for $0 \leq i \leq k + \alpha - 1$ . We also know that $r \mid a$ or $r \mid a^{a - 1} - 1$ . If we consider $r \mid a$ , because $r$ is a prime, $r = a \implies a$ is a prime number. In other words, we have just proven that if there exists a prime $r$ that divides $b!$ where $b = k + \alpha$ , then there also may exist a prime $a$ that also divides $b!$ . But because $b \geq a$ , $a = b, b - 1, ..., 1$ . Therefore, at least one of $b - j$ where $0 \leq j \leq b - 1$ , must be a prime number. Now, we have $a^{a} - a \leq b^{b} - b$ . Let us consider the highest power of prime $a$ in $b!$ . We need to find $v_a(b!)$ . By Legendre's Formula, $e_p(n!)=\sum_{i=1}^{\infty} \left\lfloor \dfrac{n}{p^i}\right\rfloor =\frac{n-S_{p}(n)}{p-1}$ . We just need to find that $\sum_{i=1}^{\infty} \left\lfloor \dfrac{b}{a^i}\right\rfloor = a$ . If $b$ is a multiple of $k$ , this will not hold true simply be plugging in. Even if $b = a + \beta$ , because the sum is finite, it would imply $\beta = 0$ which goes back to the case where $b$ is a multiple of $a$ . Therefore, the only condition for which this can hold is if $a = 1$ which contradicts the part where $1$ must divide $b - i$ because $1$ is not prime. We now have to prove that this case doesn't hold even if $r \mid a^{a - 1} - 1$ . In this case, $a$ is not necessarily prime. We can apply Fermat's Little Theorem if we assume $a - 1$ is a prime. Then, $a^{a - 1} \equiv a(\mod a - 1)$ . This tells us that $a^{a - 1} - 1 \equiv 0(\mod a - 1)$ which tells us that $a - 1$ is another prime that divides it. Therefore, there are two primes that divide $a^{a - 1} - 1$ and those are $r$ and $a - 1$ if $a - 1$ is a prime. We conclude that either $r = a - 1$ or $a^{a - 1} - 1$ has more than one prime factor. If we consider $r = a - 1$ , we can apply Legendre's again and see that it gives us the same result. If we have $a^{a - 1} - 1$ has at least two distinct prime factors and two of them are $r$ and $a - 1$ . This can only occur if $a = 2$ (The other case will lead to a - 1 being composite which is not what we assumed). But we already considered this case. Therefore, we don't have a new solution here. Now, onto when $a - 1$ is not prime.(I have not finished proving this case.)

Case 2: $b < a$

This just implies $a^{p} - p < a!$ . Because of Wilson's Theorem, if $a = k - 1$ where we use $k$ again as a prime, then $a^{p} - p$ will have a modular residue of anything but $-1 (\mod k)$ .(I have not finished proving why $b < a$ will also host no solutions).

~ilikemath247365

@@ Line 102: / Line 102: @@
 Case 1.3: <math>b</math> is anything but <math>k - 1</math> where <math>k</math> is a prime
-Recall that we have <math>a^{a} - a = b!</math>. Notice that <math>b = k</math> will lead to a contradiction because <math>a \geq 3 \implies a^{a} - a</math> is composite. Therefore, <math>b = k + \alpha</math> where <math>\alpha</math> is a constant of anything but <math>-1, 0</math>. Let <math>r</math> be a prime that divides <math>(k + \alpha)!</math>. Thus, <math>r \mid a^{a} - a = a(a^{a - 1} - 1)</math>. Therefore, we know that <math>r \mid k + \alpha - i</math> for <math>0 \leq i \leq k + \alpha - 1</math>. We also know that <math>r \mid a</math> or <math>r \mid a^{a - 1} - 1</math>. If we consider <math>r \mid a</math>, because <math>r</math> is a prime, <math>r = a \implies a</math> is a prime number. In other words, we have just proven that if there exists a prime <math>r</math> that divides <math>b!</math> where <math>b = k + \alpha</math>, then there also may exist a prime <math>a</math> that also divides <math>b!</math>. But because <math>b \geq a</math>, <math>a = b, b - 1, ..., 1</math>. Therefore, at least one of <math>b - j</math> where <math>0 \leq j \leq b - 1</math>, must be a prime number. Now, we have <math>a^{a} - a \leq b^{b} - b</math>. Let us consider the highest power of prime <math>a</math> in <math>b!</math>. We need to find <math>v_a(b!)</math>. By Legendre's Formula, <math>\[v_a(b!)=\sum_{i=1}^{\infty} \left\lfloor \dfrac{b}{a^i}\right\rfloor</math>. We just need to find that <math>\sum_{i=1}^{\infty} \left\lfloor \dfrac{b}{a^i}\right\rfloor = a</math>. If <math>b</math> is a multiple of <math>k</math>, this will not hold true simply be plugging in. Even if <math>b = a + \beta</math>, because the sum is finite, it would imply <math>\beta = 0</math> which goes back to the case where <math>b</math> is a multiple of <math>a</math>. Therefore, the only condition for which this can hold is if <math>a = 1</math> which contradicts the part where <math>1</math> must divide <math>b - i</math> because <math>1</math> is not prime. We now have to prove that this case doesn't hold even if <math>r \mid a^{a - 1} - 1</math>. In this case, <math>a</math> is not necessarily prime. We can apply Fermat's Little Theorem if we assume <math>a - 1</math> is a prime. Then, <math>a^{a - 1} \equiv a(\mod a - 1)</math>. This tells us that <math>a^{a - 1} - 1 \equiv 0(\mod a - 1)</math> which tells us that <math>a - 1</math> is another prime that divides it. Therefore, there are two primes that divide <math>a^{a - 1} - 1</math> and those are <math>r</math> and <math>a - 1</math> if <math>a - 1</math> is a prime. We conclude that either <math>r = a - 1</math> or <math>a^{a - 1} - 1</math> has more than one prime factor. If we consider <math>r = a - 1</math>, we can apply Legendre's again and see that it gives us the same result. If we have <math>a^{a - 1} - 1</math> has at least two distinct prime factors and two of them are <math>r</math> and <math>a - 1</math>. This can only occur if <math>a = 2</math>(The other case will lead to a - 1 being composite which is not what we assumed). But we already considered this case. Therefore, we don't have a new solution here. Now, onto when <math>a - 1</math> is not prime.(I have not finished proving this case.)
+Recall that we have <math>a^{a} - a = b!</math>. Notice that <math>b = k</math> will lead to a contradiction because <math>a \geq 3 \implies a^{a} - a</math> is composite. Therefore, <math>b = k + \alpha</math> where <math>\alpha</math> is a constant of anything but <math>-1, 0</math>. Let <math>r</math> be a prime that divides <math>(k + \alpha)!</math>. Thus, <math>r \mid a^{a} - a = a(a^{a - 1} - 1)</math>. Therefore, we know that <math>r \mid k + \alpha - i</math> for <math>0 \leq i \leq k + \alpha - 1</math>. We also know that <math>r \mid a</math> or <math>r \mid a^{a - 1} - 1</math>. If we consider <math>r \mid a</math>, because <math>r</math> is a prime, <math>r = a \implies a</math> is a prime number. In other words, we have just proven that if there exists a prime <math>r</math> that divides <math>b!</math> where <math>b = k + \alpha</math>, then there also may exist a prime <math>a</math> that also divides <math>b!</math>. But because <math>b \geq a</math>, <math>a = b, b - 1, ..., 1</math>. Therefore, at least one of <math>b - j</math> where <math>0 \leq j \leq b - 1</math>, must be a prime number. Now, we have <math>a^{a} - a \leq b^{b} - b</math>. Let us consider the highest power of prime <math>a</math> in <math>b!</math>. We need to find <math>v_a(b!)</math>. By Legendre's Formula, <cmath>e_p(n!)=\sum_{i=1}^{\infty} \left\lfloor \dfrac{n}{p^i}\right\rfloor =\frac{n-S_{p}(n)}{p-1}</cmath>. We just need to find that <math>\sum_{i=1}^{\infty} \left\lfloor \dfrac{b}{a^i}\right\rfloor = a</math>. If <math>b</math> is a multiple of <math>k</math>, this will not hold true simply be plugging in. Even if <math>b = a + \beta</math>, because the sum is finite, it would imply <math>\beta = 0</math> which goes back to the case where <math>b</math> is a multiple of <math>a</math>. Therefore, the only condition for which this can hold is if <math>a = 1</math> which contradicts the part where <math>1</math> must divide <math>b - i</math> because <math>1</math> is not prime. We now have to prove that this case doesn't hold even if <math>r \mid a^{a - 1} - 1</math>. In this case, <math>a</math> is not necessarily prime. We can apply Fermat's Little Theorem if we assume <math>a - 1</math> is a prime. Then, <math>a^{a - 1} \equiv a(\mod a - 1)</math>. This tells us that <math>a^{a - 1} - 1 \equiv 0(\mod a - 1)</math> which tells us that <math>a - 1</math> is another prime that divides it. Therefore, there are two primes that divide <math>a^{a - 1} - 1</math> and those are <math>r</math> and <math>a - 1</math> if <math>a - 1</math> is a prime. We conclude that either <math>r = a - 1</math> or <math>a^{a - 1} - 1</math> has more than one prime factor. If we consider <math>r = a - 1</math>, we can apply Legendre's again and see that it gives us the same result. If we have <math>a^{a - 1} - 1</math> has at least two distinct prime factors and two of them are <math>r</math> and <math>a - 1</math>. This can only occur if <math>a = 2</math>(The other case will lead to a - 1 being composite which is not what we assumed). But we already considered this case. Therefore, we don't have a new solution here. Now, onto when <math>a - 1</math> is not prime.(I have not finished proving this case.)
 Case 2: <math>b < a</math>

2022 IMO (Problems) • Resources
Preceded by Problem 4	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 6
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