2008 USAMO Problems/Problem 1

From AoPSWiki

Problem

(Titu Andreescu) Prove that for each positive integer $n$ , there are pairwise relatively prime integers $k_0, k_1 \dotsc, k_n$ , all strictly greater than 1, such that $k_0 k_1 \dotsm k_n -1$ is the product of two consecutive integers.

Solutions

Solution 1

We will prove the problem for each nonnegative integer $n$ . We wish to show that $k_0 k_1 \dotsc k_n = a^2+a+1,$ for some integer $a$ . We induct on $n$ . For our base case, $n=0$ , we may let $a$ be positive integer.

For the inductive step, suppose that $k_0, \dotsc, k_{n-1}$ are pairwise relatively prime integers such that $k_0 \dotsm k_{n-1} = a^2+a+1,$ for some integer $a$ . Let $k_n = a^2 + 3a+3$ . Evidently, $k_n >1$ . Also, $\gcd(a^2+a+1, k_n) = \gcd\bigl(a^2+a+1, k_n - (a^2+a+1) \bigr) = \gcd\bigl(a^2 + a+1, 2(a+1) \bigr) .$ Since $a^2+a+1 = a(a+1)+1$ is odd and relatively prime to $a+1$ , it follows that $a^2+a+1$ and $k_n$ are relatively prime, so $k_n$ is relatively prime to each of $k_0, \dotsc, k_{n-1}$ . Finally, $\begin{align*}k_0 k_1 \dotsm k_n &= (a^2+a+1)(a^2+3a+3)\\&= \bigl[ (a+1)^2-a \bigr] \cdot \bigl[ (a+1)^2 + a+2 \bigr]...$ This completes the induction. $\blacksquare$

Solution 2

Lemma. If $p$ is prime such that $p\equiv 1 \pmod{3}$ , there exists a residue $r$ such that $p \mid r^2+r+1$ .

Proof. Let $g$ be a multiplicative generator of the nonzero integers mod 3. Set $r \equiv g^{(p-1)/3}$ . Then $r-1 \not\equiv 0 \pmod{p}$ , but $r^3-1 \equiv 0 \pmod{p}$ , so $\frac{r^3-1}{r-1} \equiv r^2 + r+1 \equiv 0 \pmod{p}$ . $\blacksquare$

By Dirichlet's Theorem, there are infinitely many primes congruent to 1 (mod 3). Let $p_0, \dotsc, p_n$ be $(n+1)$ such primes, and let $r_0, \dotsc, r_n$ be respective residues as described in the lemma. By the Chinese Remainder Theorem, there is a positive integer $a$ that satisfies the relation $a \equiv r_i \pmod{p_i}$ for each integer $i \in [0,n]$ . Then $p_0 \dotsc p_n \mid a^2+a+1 .$ Now, for $1 \le i \le n$ , take $k_i$ to be the greatest power of $p_i$ that divides $a^2 + a +1$ , and let $k_0 = (a^2+a+1)/(k_1 \dotsm k_n)$ . Since all the $k_i$ are pairwise relatively prime and are greater than 1, we are done. $\blacksquare$

Solution 3

Firstly, we see that there are $n + 1$ numbers $k_i$ . Since $n\geq1$ , there are at least 2 values of $k_i$ . Define a relatively prime partition to be a set of relatively prime numbers such that their product is equal to some natural number.

Define $P$ to be the greatest possible cardinality of a relatively prime partition for that number.

Lemma 1: All cardinalities of the relatively prime partitions of a number up to $P$ can be attained.

Proof: $P = P - 0$ satisfies the properties. For any $P - n$ which satisfies the properties, we can take any of the 2 $P - n$ numbers and multiply them together. Because they are both relatively prime to all the other numbers, their product is relatively prime to all the other numbers as well, and that results in $P - n - 1$ numbers which satisfy the conditions, unless $P - n = 1$ , because there is only one number left. Therefore, all numbers of numbers of relatively prime factors from $1$ to $P - 1$ are attainable, if $P$ is attainable as well.

End Lemma

Lemma 2: $a^2 + a + 1$ can have arbitrarily many prime factors for some value of $a$ .

Proof: Let $f(x) = x^2 + x + 1$ . Let $a^2 + a + 1 = k$ . Then $f(a + k) = a^2 + 2ak + k^2 + a + k + 1 = 2ak + k^2 + 2k$ . After factoring, we get $k(k + 2 + 2a) = (a^2 + a + 1)(a^2 + 3a + 3)$ . $a^2 + 3a + 3 \geq a^2 + a + 1$ , because $a$ is strictly positive. Let their GCD=g. $g|2a + 2$ $g|2a^2 + 2a$ $g|a^2 - 3a - 3 + a^2 + 3a + 3, g|2a^2$ $g|2a$ , and so $g|2$ , but $g|a^2 + a + 3$ which is strictly odd, so g=1. $(a^2 + a + 1)(a^2 + 3a + 3)$ therefore must contain prime factors not in $a^2 + a + 1$ . Upon repeating this an arbitrary number of times, we get a number of the form $a^2 + a + 1$ which has arbitrarily many distinct prime factors.

End Lemma

Then $a(a + 1) + 1 = a^2 + a + 1$ would not have $n + 1 = P$ elements in the relatively prime partition for some value of n, and any value of a. It is possible to choose a value of $a$ such that $a^2 + a + 1$ has arbitrarily many unique prime factors, by our second Lemma, and so it is possible for $P$ to be arbitrarily high. By our first Lemma, all numbers up to P are possible values for the cardinality of some relatively prime partition, and so there always exists some number that has an arbitrary number of elements in a relatively prime partition. Because $a^2 + a + 1 = a(a + 1) + 1$ , $a(a + 1)$ is the product of 2 consecutive integers, we see that the given statement is true because if $k_0k_1...k_n = a(a + 1) + 1$ , then $k_0k_1...k_n - 1 = a(a + 1)$ .