Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...
Can you explain why a sequence of operations always gives you perfect squares?
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you
notice when successive terms are taken? What happens to the terms
if the fraction goes on indefinitely?
This is the solution sent in by Yatir Halevi. Thanks Yatir. A
correct solution was also received from Andrei Lazanu.
Let's say we want to find the square of $a$
We know that $a^2 = a^2-b^2+b^2 = (a+b)\times(a-b)+b^2$and for
every a, we can pick a certain b that will make the
calculation$a^2$ as easy as possible.
So, if$a$ is a number that ends with a 5: it can be written as
$$a=10\times q +
So $a^2$is equal to $q(q+1)$ plus two zeros after it $(10^2)$ that
are "stolen" by the 25 that is added on.