Square Triangular NumbersDate: 11/22/2002 at 14:24:27 From: Sharon Subject: Perfect square Is there a math equation for a perfect square, using the following as a definition for the perfect square with the following properties? The sum of the integers is equal to the number squared. Example: 1+2+3+4+5+6+7+8=36 36 = 6 squared, making 36 a perfect square Another example would be 1225 1225 = sum of 1 to 49 1225 = 35 squared, making 1225 a perfect square Date: 11/22/2002 at 15:00:25 From: Doctor Roy Subject: Re: Perfect square Hi, Thanks for writing to Dr. Math. You are talking about numbers that are both triangular and square, or square triangular numbers. You already know what a square number is. A triangular number is a number that can be expressed as the sum of the first n whole numbers, like 1, 3, 6, 10, 15, 21, ..... This problem was solved a while ago. Here's the solution: (1 + 2 + 3 + .... + n) = n*(n+1)/2 This is the n-th triangular number. Let's say it is the same as the mth square, m^2, or: n*(n+1)/2 = m^2 Let's complete the square on the left: 1/2 * (n^2 + n + (1/2)^2) - (1/2)*(1/2^2) = m^2 1/2 * (n + 1/2)^2 - 1/8 = m^2 4*(n+1/2)^2 - 8*m^2 = 1 (2n+1)^2 - 8m^2 = 1 From here, let's substitute x = (2n+1) and y = 2m x^2 - 2*y^2 = 1 This is a famous equation known as the Pell equation. The solutions are well known. The first few solutions you know: m = 6, n = 8 and m = 35, n = 49. A few others are: 41616, 1413721, 48024900 These numbers are both triangular and square. Euler proved that there is an infinite number of such numbers. There is an abbreviated proof in the Dr. Math archives at: Triangular Numbers http://mathforum.org/library/drmath/view/51528.html There is a general formula for square triangular numbers. You can find it by searching the Mathworld site for square triangular numbers. The Mathworld website is at: http://mathworld.wolfram.com I hope this helps. - Doctor Roy, The Math Forum http://mathforum.org/dr.math/ |
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