published her prize-winning essay at her own expense in 1821, mostly because she wanted to present her work in opposition to that of Poisson. In the essay she pointed out some of the errors in his method. In 1826 she submitted a revised version of her 1821 essay to the Academy. According to Andrea Del Centina, the revision included attempts to clarify her work by "introducing certain simplifying hypotheses". This put the Academy in an awkward position, as they felt the paper to be "inadequate and trivial", but they did not want to "treat her as a professional colleague, as they would any man, by simply rejecting the work". So Augustin-Louis Cauchy, who had been appointed to review her work, recommended her to publish it, and she followed his advice. One further work of Germain's on elasticity was published posthumously in 1831, her "". She used the mean curvature in her research (see Honors in number theory). Later work in number theory Renewed interest Germain's best work was in number theory, and her most significant contribution to number theory dealt with Fermat's Last Theorem. In 1815, after the elasticity contest, the Academy offered a prize for a proof of Fermat's Last Theorem. It reawakened Germain's interest in number theory, and she wrote to Gauss again after ten years of no correspondence. In the letter, Germain said that number theory was her preferred field and that it was in her mind all the time she was studying elasticity. She outlined a strategy for a general proof of Fermat's Last Theorem, including a proof for a special case. Germain's letter to Gauss contained her substantial progress toward a proof. She asked Gauss whether her approach to the theorem was worth pursuing. Gauss never answered. Her work on Fermat's Last Theorem Fermat's Last Theorem can be divided into two cases. Case 1 involves all powers p that do not divide any of x, y, or z. Case 2 includes all p that divide at least one of x, y, or z. Germain proposed the following, commonly called "Sophie Germain's theorem": Let p be an odd prime. If there exists an auxiliary prime P = 2Np + 1 (N is any positive integer not divisible by 3) such that: if xp + yp + zp ≡ 0 (mod P), then P divides xyz, and p is not a p-th power residue (mod P). Then