N rem R

choose two digits (e.g. 2 and 5)

• divide the sum of their squares by their sum (i.e. (4 + 25) divided by 7 for the example)
• write this as a number and a remainder: (i.e. 29 / 7 = 4 rem 1)

try a few of these

something special happens for two consecutive numbers
what is it?

a proof of this property involves (for n and (n + 1)):
splitting the numerator into 2n^2 + n + (n + 1)
dividing by (2n + 1)
that gives n  +  (n + 1)/(2n + 1)
which is  n  rem  (n + 1)