Department of Mathematics
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George Orwell (in "1984"): 'Sometimes, Winston, Two and two are five. You must try harder. It is not easy to become sane'.

Let \(k\) be an infinite field. A formal group law \( x\oplus y = f(x, y) \), \( f(x, y)\in k(x, y) \) is called Orwellian if \(2\oplus 2 = 5\).

In this talk, we show how the theorems of Nagell-Ramanujan and J.L. Mordell imply the existence of several Orwellian group laws, thus giving examples to George Orwell's claim in his now famous dystopian novel "1984". In fact, we show that there are infinitely many such group laws. Being one-dimensional formal groups, these group laws must be either addition or multiplication (see, e.g. [1]). We give an elementary argument to show that these Orwellian groups are all isomorphic to the multiplicative group \( (k^{*};\, \cdot)\).

This is a joint work done in collaboration with Dr. Alok Shukla.

References:

1. Coleman, Robert F.; McGuinness, Francis Oisín. Rational formal group laws. Pacific J. Math. 147 (1991), no. 1, 25–27.
2. L.J. Mordell. The diophantine equations \(x^2 + 7 = 2n\). Ark. Mat 4(1962), 455–460
3. T Nagell. The Diophantine equation \(x^2 + 7 = 2n\). Arkiv för Matematik, 4(2):185–187,1961.
4. George Orwell, Nineteen eighty-four, Secker & Warburg, London, 1949.