So I thought I would tell you a little story about how I was inspired by DrMaths (Steve Humble) to do some maths on my fridge last Friday after work.
Steve has been tweeting number facts as @DrMaths for some time and last Friday's was
If you square 1, 5 and 6 you get 62. There are three different numbers you can square to get 62! Its the smallest number with this propertyBy which I thought he meant there were three sets of three numbers that when squared would add up to 62.
Almost immediately I figured out in my head that 2^2+3^2+7^2=62, but for the life of me I couldn't work out another way.
So I got out my board marker and sat on my kitchen floor and started doing maths on my fridge. You can see photographic evidence of the results below. I am certain that I've proved (by brute force) that there are only 2 ways to this answer and the smallest number with three sets of three summed square numbers is 101.
I wish to add here that DrMaths did clarify in a later tweet to someone who asked about it
too few words used badly! "two distinct squares"I then went a bit further and started working out prime numbers and decided that 1^2+2^2+6^2=41 is the smallest one of these "sum of 3 squares" numbers that is prime. I then wondered which would be the smallest prime that had two distinct sums. It seems the answer is 89.
While of course 101 is both the smallest integer and the smallest prime number with three distinct sums of this form.