106.20 When do we have 1 + 1 = 11 and 2 + 2 = 5?

Abstract

The phrase "$ 2+2=5 $" is a clich\'e or a slogan used in political speeches, propaganda, or literature, most notably in the novel "$ 1984 $" by George Orwell. More recently, we came across a You-Tube short film comedy, Alternative Math, produced by IdeaMan Studios (see \cite{Danny}). It is a hilarious exaggeration of a teacher who is dragged through the mud for teaching that $ 2+2=4 $ and not $ 22 $, as Danny, a young student, kept on insisting. In the movie, Danny and the whole community sincerely believe $ 1+1=11 $ and $ 2+2=22 $. Jokes aside, we ask the question whether a polynomially defined group law "$\oplus$" defined over the field of rationals such that $ 1\oplus1=u $ and $ 2 \oplus2=v $ can simultaneously be satisfied for arbitrary integers $u$ and $ v $. Answer to this question takes us through a fascinating journey from Brahmagupta all the way to the modern works of Louis Joel Mordell and Ramanujan!