My 1st Putnam Problem

Prove that there exist integers $a, b, c$, not all zero, and each of absolute value less than \(10^6\), such that \[|a+b\sqrt{2}+c\sqrt{3}|<10^{-11}\] Proof: We are given that the range of \(a, b, c\) are \(-10^6+1\leq a, b, c \leq 10^6-1\), so the weaker result, \(-\frac{10^6}{2}\leq a, b, c \leq \frac{10^6}{2}\) is also true. This gives us a total of \((10^6+1)^3\) possible values for \(a+b\sqrt{2}+c\sqrt{3}\).

Since \(10^6+1>10^6\), squaring both sides gives \((10^6+1)^3 > 10^{18}\) or \((10^6+1)^3 \geq 10^{18}+1\), because the \(``>"\) implies a ceiling function on the integers.

Note that since \(a+b\sqrt{2}+c\sqrt{3}\) is linear in \(a, b, c\), we can find the range of all its values by minimizing and maximizing the values of \(a, b, c\). Substituting in \((-10^6+1, -10^6+1, -10^6+1)\) for the minimum and \((10^6-1, 10^6-1, 10^6-1)\) gives us the following range

\[(-10^6+1)(1+\sqrt{2}+\sqrt{3})\leq a+b\sqrt{2}+c\sqrt{3} \leq (10^6-1)(1+\sqrt{2}+\sqrt{3})\]

Thus, the size of the range of possible values of \(a+b\sqrt{2}+c\sqrt{3}\) is \((1+\sqrt{2}+\sqrt{3})(2\cdot10^6-2)\). Simple calculation finds that \(1+\sqrt{2}+\sqrt{3}<5\), so \((1+\sqrt{2}+\sqrt{3})(2\cdot10^6-2)<10^7-10\).

If we divide our range, \(10^7-10\), into \(10^{18}\) sections of equal length, each section has length \(\frac{10^7-10}{10^{18}}<10^{-11}\). Although we have more than \(10^{18}+1\) possible values for the value of \(a+b\sqrt{2}+c\sqrt{3}\), we will use the values where \(\frac{10^6}{2}\leq a, b, c \leq \frac{10^6}{2}\).

Since there are \(10^{18}+1\) possible values of \(a+b\sqrt{2}+c\sqrt{3}\) and \(10^{18}\) segments, distributing the values across all segments guarantees that two different sums will lie in the same segment by Pigs-In-Holes Principle. Since they lie in the same segment of length less than \(10^{-11}\), their absolute difference will be less than \(10^{-11}\) and we can express their difference in the form \((a_1-a_2)+(b_1-b_2)\sqrt{2}+(c_1-c_2)\sqrt{3}\). Since we took \(a_1, a_2, b_1, b_2, c_1, c_2\) from the range \([-\frac{10^6}{2}, \frac{10^6}{2}]\), their differences must satisfy \(10^6 < a_1-a_2, b_1-b_2, c_1-c_2 < 10^6\).