2Problems Leading to Pell’s Equationand Preliminary InvestigationsThe first chapter presented a situation that led to pairs of integers (x, y) that satisfiedequations of the form x2−2y2 k for some constant k. One of the reasons for thepopularity of Pell’s equation as a topic for mathematical investigation is the factthat many natural questions that one might ask about integers lead to a quadraticequation in two variables, which in turn can be cast as a Pell’s equation. In thischapter we will present a selection of such problems for you to sample.For each of these you should set up the requisite equation and then try to findnumerical solutions. Often, you should have little difficulty in determining at leastone and may be able to find several. These exercises should help you gain someexperience in handling Pell’s equation. Before going on to study more systematicmethods of solving them, spend a little bit of time trying to develop your ownmethods.While a coherent theory for obtaining and describing the solutions of Pell’sequation did not appear until the eighteenth century, the equation was tackledingeniously by earlier mathematicians, in particular those of India. In the thirdsection, inspired by their methods, we will try to solve Pell’s equation.2.1 Square and Triangular NumbersThe numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, ... , tn≡12n(n + 1),... arecalled triangular, since the nth number counts the number of dots in an equilateraltriangular array with n dots to the side.It is not difficult to see that the sum of two adjacent triangular numbers is square.Figure 2.1.2.1. Square and Triangular Numbers 17Figure 2.2.But does it often happen that an individual triangular number is square? We willexamine this and similar questions.Exercise 1.1. Verify that the condition that the nth triangular number tnis equalto the mth square is that12n(n + 1) m2. Manipulate this equation into the form(2n + 1)2− 8m2 1.Thus, we are led to solving the equation x2− 8y2 1 for integers x and y.It is clear that for any solution, x must be odd (why?), so that we can then findthe appropriate values of m and n. Observe that 1 and 36 are included in the listof triangular numbers. What are the corresponding values of x, y, m, n? Use theresults of Exercise 1.1.7(c) to generate other solutions.Exercise 1.2. There are triangular numbers that differ from a square by 1, suchas 3 22− 1, 10 32+ 1, 15 42− 1, and 120 112− 1. Determine otherexamples.Exercise 1.3. Find four sets of three consecutive triangular numbers whoseproduct is a perfect square.Exercise 1.4. Find four sets of three consecutive triangular numbers that add upto a perfect square.Exercise 1.5. Determine integers n for which there exists an integer m for which1 + 2 + 3 +···+m (m + 1) + (m + 2) +···+n.Exercise 1.6. Determine positive integers m and n for whichm + (m + 1) +···+(n − 1) + n mn(International Mathematical Talent Search 2/31).18 2. Problems Leading to Pell’s Equation and Preliminary InvestigationsExploration 2.1. The triangular numbers are sums of arithmetic progressions. Wecan ask similar questions about other arithmetic progressions as well. Determinethe smallest four values of n for which the sum of n terms of the arithmetic series1 +5 +9 +13 +···is a perfect square. Compare these values of n with the termsof the sequence {qn} listed in Exploration 1.1. Experiment with other initial termsand common differences.Exploration 2.2. Numbers of the form n(n + 1) (twice the triangular numbers)are known as oblong, since they represent the area of a rectangle whose sideslengths are consecutive integers. The smallest oblong numbers are2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156.A little experimentation confirms that the product of two consecutive oblong num-bers is oblong; can you give a general proof of this result? Look for triples (a,b,c)of oblong numbers a, b, c for which c ab. For each possible value of a,investigate which pairs (b, c) are possible.An interesting phenomenon is the appearance of related triples of solutions. Forexample, we have (a,b,c)equal to(14 × 15, 782 × 783, 11339 × 11340),(14 × 15, 13 × 14, 195 × 196),(13 × 14, 782 × 783, 10556 × 10557),while(11339 × 11340)(13 × 14)2 (195 × 196)(10556 × 10557).Are there other such triples?2.2 Other Examples Leading to a Pell’s EquationThe following exercises also involve Pell’s equation. For integers n and k with1 ≤ k ≤ n, we define nk n(n − 1) ···(n − k + 1)1 · 2 ···k n!k!(n − k)!.Also, we definen0 1 for each positive integer n. Observe that 1 +2 +···+n n+12.Exercise 2.1. Determine nonnegative integers a and b for which ab a − 1b + 1.2.2. Other Examples Leading to a Pell’s Equation 19Exercise 2.2. Suppose that there are n marbles in a jar with r of them are redand n − r blue. Two marbles are drawn at random (without replacement). Theprobability that both have the same color is12. What are the possible values of nand r?Exercise 2.3. The following problem appeared in the American MathematicalMonthly (#10238, 99 (1992), 674):(a) Show that there exist infinitely many positive integers a such that both a + 1and 3a + 1 are perfect squares.(b) Let {an}be the increasing sequence of all solutions in (a). Show that anan+1+1is also a perfect square.Exercise 2.4. Determine positive integers b for which the number (111 ...1)bwith k digits all equal to 1 when written to base b is a triangular number, regardlessof the value of k.Exercise 2.5. Problem 2185 in Crux Mathematicorum (22 (1996), 319) points outthat22+ 42+ 62+ 82+ 102 4 · 5 + 5 · 6 + 6 · 7 + 7 · 8 + 8 · 9and asks for other examples for which the sum of the first n even squares is thesum of n consecutive products of pairs of adjacent integers.Exercise 2.6. Determine integer solutions of the system2uv − xy 16,xv − uy 12(American Mathematical Monthly 61 (1954), 126; 62 (1955), 263).Exercise 2.7. Problem 605 in the College Mathematics Journal (28 (1997), 232)asks for positive integer quadruples (x,y,z,w)satisfying x2+ y2+ z2 w2for which, in addition, x y and z x ± 1. Some examples are (2, 2, 1, 3) and(6, 6, 7, 11). Find others.Exercise 2.8. The root-mean-square of a set {a1,a2,...,ak} of positive integersis equal toa21+ a22+···+a2kk.Is the root-mean-square of the first n positive integers ever an integer? (USAMO,1986)Exercise 2.9. Observe that (1 + 12)(1 + 22) (1 + 32). Find other examples