May 2025
This month’s problem is inspired by Question 9 on the 2025 Euclid contest. The question in its original form is on the next page. As a warm up, try the problem yourself before attempting the Problem of the Month.
While solving Q9(c), you need to find pairs of integers \((m,e)\) satisfying that \(m^2 - 8e^2\) is a perfect square. Let’s investigate this problem further.
For each of the following equations, find one pair \((m,e)\) of non-zero integers that solve it: \(m^2 - 8e^2 = 1\), \(m^2 - 8e^2 = 4\) and \(m^2 - 8e^2 = 9\).
Integers \(a\) and \(b\) are called coprime if they share no positive common divisors other than \(1\). For example, \(3\) and \(5\) are coprime but \(4\) and \(6\) are not.
Find a pair \((m,e)\) of coprime integers satisfying \(m^2 - 8e^2 = 49\).
Let’s focus on expressions of the form \(a + b\sqrt 8\), where \(a\) and \(b\) are integers. Define the norm of \(a + b\sqrt 8\) to be \(N(a + b\sqrt 8) = (a+b\sqrt 8)(a - b\sqrt 8)\).
Let \(a,b,c,d\) be integers. Prove that \(N\big((a + b\sqrt8)(c + d\sqrt 8)\big) = N(a + b\sqrt8)N(c + d \sqrt 8)\).
It turns out that \(19^2 - 8(3^2) = 17^2\) and \(27^2 - 8(5^2) = 23^2\). Find coprime integers \(a,b\) so that \(a^2 - 8b^2 = 391^2\).
Find infinite sequences of integers \(a_1,a_2,\ldots\) and \(b_1,b_2,\ldots\) satisfying that for all positive integers \(n\),
\(a_n\) and \(b_n\) are coprime, and
\(a_n^2 - 8b_n^2 = 7^{2^n}\).
When we write \(7^{2^n}\) we mean \(7^{(2^n)}\). So, for example, when \(n = 5\), \(7^{2^n}\) is equal to \(7^{32}\) and not \(49^5\).
Here is Question 9 from the 2025 Euclid contest.
Suppose that \(p(x) = qx^3 - rx^2 - sx + t\) for some positive integers \(q<r<s<t\) which form an arithmetic sequence.
Show that \(x = 1\) is a root of \(p(x)\).
Suppose that the average of \(q\), \(r\), \(s\), \(t\) is 19 and that \(p(x)\) has three rational roots. Determine the roots of \(p(x)\).
Prove that, for every positive integer \(n > 3\), there are at least two arithmetic sequences of positive integers \(q<r<s<t\) with common difference \(2n\) for which \(p(x)\) has three rational roots.
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant, called the common difference. For example, \(3, 5, 7, 9\) are the first four terms of an arithmetic sequence.)