Taxicab Numbers Extended
Taxicab numbers are numbers that can be written in two distinct ways as the sum of two cubes, the smallest of which is 1729. (1729 = 1^3 + 12^3 = 9^3 + 10^3.)
I decided it would be a good idea to try and find numbers that can be written as the sum of two squares in two distinct ways, and so far I have found 18 of them. In ascending order,
50, 65, 85, 125, 130, 145, 170, 185, 205, 250, 260, 265, 290, 305, 325, 340, 365, 370.
Unfortunately, I haven't determined a uniform way to find these numbers- so, I've just been checking all the multiples of 5. (I checked every number between 1 and 100, and noticing that numbers that satisfy k = 0 mod 5 are the only ones that work, I only checked multiples of 5 from then on.) So, there could be some missing in between. (There might even be some multiples of 5 I missed in the 300's because I kind of got lazy and started writing down the ones that worked and not checking the ones I was pretty sure didn't work.) I also have no idea why only numbers that are multiples of 5 would work, but I have a vague idea.
If we look at the digit that ends up in the digits place when you square the 10 different 1 digit numbers, we get 0, 1, 4, 9, 6, 5, 6, 9, 4, 1. There are 6 digits you can end up with, namely 0, 1, 4, 5, 6 and 9. Combining these, you can make any digit. However, most can only be made with one or two combinations of numbers- 5 and 0 can be made with 3 different combinations. (5 = 1+4 = 0+5 = 9+6, and 0 = 0+0 = 1+9 = 4+6.) This makes these two digits the most likely to be in the digits place, but it is no help in proving that they will be the only digits in the digits place of my numbers.
Another more obscure pattern that I noticed is in the numbers that you square to create my numbers. For example, 50 = 5^2 + 5^2 = 7^2 + 1^2. Notice that 5+5 = 10 is 2 greater than 7+1 = 8. In other words, for a number k that can be written as both a^2 + b^2 and c^2 + d^2 for positive integers a, b, c and d and for a+b > c+d, a+b-c-d = 2. However, this only holds for the first 5- after that, the difference jumps to either 4 or 6 or back to 2. Although less specific, another rule can form- perhaps a+b-c-d = 0 mod 2.
The numbers seem to show up almost randomly, and the problem is a relatively unfamiliar one so I'm not sure where to go from here.
