Math 341: Number Theory Section 2.1: Polygonal Numbers Mckenzie West Last Updated: February 21, 2021 We acknowledge that UW-Eau Claire occupies the sacred and ancestral lands of Indigenous Peoples. We honor the land of the Ojibwe and Dakota Nations. 2 Last Time. { The Binomial Theorem { Pascal's Triangle { Properties of Binomial Coefficients Today. { Triangular Numbers { Pentagonal Numbers 3 Definition. Recall the definition of the nth triangular number, n(n + 1) T = 1 + 2 + 3 + ··· + n = : n 2 Similarly, we call m 2 N triangular if there is some n for which m = Tn. Exercise 1. n(n + 1) n + 1 Verify that = . 2 2 4 Exercise 2. What are the first 10 triangular numbers? n 1 2 3 4 5 6 7 8 9 10 Tn 111 111 111 111 111 111 111 111 111 111 5 Exercise 3. Go to https://oeis.org/A000217 to see a longer list of triangular numbers. (a) Can you find two triangular numbers whose sum and difference are also triangular? (b) How about three consecutive triangular numbers whose product is a perfect square? (c) How about three consecutive triangular numbers whose sum is a perfect square? 6 Exercise 4. Show that for all n ≥ 1, 8Tn + 1 is a perfect square. 7 Exercise 5. Follow the steps below to show that if 8m + 1 = n2 for some nonzero n 2 Z, then m is triangular. (Do write your work on your own paper, this will be hard to write entirely out on the slide.) (a) Explain why n must be odd. n−1 (b) Let k = 2 . What is Tk ? 2 (c) Verify that 8Tk + 1 = n . (d) Conclude that m = Tk (why), so m is triangular. 8 Exercise 6. Let n be a triangular number. Show that 9n + 1 is also triangular. k(k+1) (a) Let n = Tk = 2 . j(j+1) (b) Find the integer j 2 N such that 2 = 9Tk + 1. (c) Conclude that 9n + 1 is triangular.(WHY??) Example. 9 Here's how we would write the formal proof for the fact that if n is triangular then so is 9n + 1. Proof. Let n 2 N be a triangular number. We want to show 9n + 1 is also triangular. Since n is triangular, there is a k 2 N such that k + 1 k(k + 1) n = T = = : k 2 2 Via algebraic manipulation, we see k(k + 1) 9k2 + 9k + 2 (3k + 1)(3k + 2) 9n + 1 = 9 + 1 = = : 2 2 2 Let j = 3k + 1 2 N. Substituting this value, (3k + 1)(3k + 2) j(j + 1) 9n + 1 = = = T : 2 2 j Therefore, we conclude that 9n + 1 = Tj is also triangular. 10 Exercise 7. Show that the difference between the squares of two consecutive triangular numbers is always a cube. 11 Exercise 8. Prove that for all n, 1 1 1 1 1 + + + + ··· + < 2: 1 3 6 10 tn 2 1 1 (Hint: Observe that n(n+1) = 2( n − n+1 ).) (a) Use the hint to find an algebraic formula for 1 1 1 1 1 + + + + ··· + : 1 3 6 10 tn (b) Verify that the formula you found will be less than 2 no matter what the value of n 2 N. Definition. 12 Pentagonal numbers are integers that represent the number of dots that can be arranged evenly in a pentagon: In general, p1 = 1 and pn = pn−1 + (3n − 2) for n ≥ 2. Exercise 9. What are the first 10 pentagonal numbers? n 1 2 3 4 5 6 7 8 9 10 111 111 111 111 111 pn 1 5 12 22 35 111 111 111 111 111 13 Exercise 10. Use induction, and the inductive definitions: p1 = 1 and pn = pn−1 + (3n − 2), to show that n(3n − 1) p = ; n ≥ 1: n 2 14 Exercise 11. For n ≥ 2, verify 2 pn = tn−1 + n : 15 Exercise 12. Show that for all n ≥ 2, 2tn−1 + tn is a pentagonal number. 16 Definition. The generating function for a sequence a1; a2; a3;::: is the formal power series 1 2 3 X n a1x + a2x + a3x + ··· = anx : n=1 Note. Generating functions don't have to start at n = 1, but this one does. We call this a \formal power series" because we have no intention of plugging in a value for x. Instead, we use x as a place marker. 17 Exercise 13. Write out the first few terms of the generating function for Tn, the triangular numbers, and for pn, the pentagonal numbers. 1 X n (a) Tnx = n=1 1 X n (b) pnx = n=1 18 Recall. 1 2 3 In Calculus II, you learned 1−x = 1 + x + x + x + ··· ; for all x 2 R with jxj < 1. Exercise 14. I claim that the generating function for Tn satisfies 1 X x T xn = : n (1 − x)3 n=1 Verify that the coefficients of x1; x2; x3; x4 do match. Here's the right-hand side, re-written using the Calc II identity. x = x(1+x+x2+x3+··· )(1+x+x2+x3+··· )(1+x+x2+x3+··· ) (1 − x)3.