Square Sum Concatenation - Number Theory 3/4 Challenge
This challenge was set by Dr Herbert Gangl during one of his Number Theory 3/4 lectures.
Observe that \( 12^2 + 33^2 = 1233 \), find as many such pairs \( (x,y) \), with \( x^2 + y^2 = x \text{ concat } y \), as possible.
- Read a brief explanation of how I found these results.
- View some slides from a presentation I gave about this in the last Number Theory 3/4 problems class.
This article by van der Poorten, Thomsen and Wiebe contains an alternative explanation, a way generating examples, and some other cute observations. (And a more appealing name: self-similar sums of squares.)
Solutions are hosted on github, with direct links below
Solutions with \( y \) having the given number of digits:
- Solutions 1-46 digits
- Solutions 47-100 digits ~9MB (excluding some which remain uncalculated, and 75)
- Solutions 75 digits ~35MB
- Solutions from non-squarefree factorisations
Families generated by solution where \( y \) has given number of digits. Note: there is (probably a lot of) repetition of families. When prepending the repeating block to the initial solution, the initial solution may need \( x \) to be padded with a \( 0 \) to make \( x \), and \( y \) the same length, similarly for the repeating block.
- Families 1-46 digits
- Families 47-100 digits (excluding some which remain uncalculated, and 75)
- Families 75 digits
- Families from non-squarefree factorisations