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Secondary Magazine - Issue 37: Focus on

This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 23 June 2009 by ncetm_administrator
Updated on 06 July 2009 by ncetm_administrator

Secondary Magazine Issue 37square numbers squares

Focus on...square numbers

The nth square number can be found by squaring n and also by summing the odd numbers up to the nth odd number.

An image to help see why summing odd numbers generates a square number can be seen below:

summing odd numbers generates a square number

Summing two consecutive triangle numbers also generates a square number:

 summing two consecutive triangle numbers also generates a square number

A square number can only end with digits 00, 1, 4, 6, 9 or 25. To square a number that ends in a 5, multiply (the digits before the 5) by (the digits before the 5 +1). This gives all of the digits of the square except for the tens and units column which will be 25. For example, to calculate 852 multiply 8 by (8+1) which gives 72. So 852 = 7225. 

In trying to prove Goldbach’s Conjecture, Chinese mathematician Chen Jingrun has proved that there always exists a number which is either a prime or the product of two primes between any two consecutive square numbers. This is close to Legendre’s conjecture, proposed by Adrien-Marie Legendre, which states that there is a prime number between any two consecutive square numbers. Legendre’s theorem is currently unproven and is one of Landau’s Problems from 1912.
As a part of the study of Waring's Problem, it has been shown that every positive integer is the sum of no more than 4 positive squares. This Java applet will break any integer into its square components.
The smallest square number to contain all of the digits 1 to 9 once is 118262  = 139854276, the largest is 303842 = 923187456.
Joseph Madachy, the longtime editor of Journal of Recreational Mathematics, explored numbers that are equal to the sum of the squares of their two "halves" such as:
122 + 332 = 1233
882 + 332 = 8833
102 + 1002 = 10100
5882 + 23532 = 5882353
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