## Fields Satisfying 1+1=11 and 2+2=22

Date Friday, May 24, 2019

While browsing through funny math vides, I came across a YouTube short film comedy, Alternative Math, produced by IdeaMan Studios (see [3]). 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 sincerely believes 1+1 = 11 and 2+2 = 22. Now the addition law "x + y" is a binary associative polynomial over a field. Here we ask for examples of fields k admitting "genuine" associative additions ⊕ where both equations 1⊕1=11 and 2⊕2=22 are valid. Indeed there are infinitely many such fields, both finite and infinite. In this talk we characterize all fields having a polynomially defined group law ⊕ in k[x, y] satisfying both 1⊕1=11 and 2⊕2=22. We also give an example of a non-polynomial rational function x⊕y ε Q(x, y), the field of binary rational functions over Q and satisfying the two Danny properties.

References:

[1] Brawley, Joel V; Gao, Shuhong; Mills, Donald., Finite fields and applications (Augsburg, 1999), 43–56, Springer, Berlin, 2001.

[2] Padmanabhan, R., Group laws satisfying 1+1=11 and 2+2=22, Resonance, vol 24, Springer 2019, to appear.