$a^{4} + 4b^{4}=(a^2+2b^2+2ab)(a^2+2b^2-2ab)$
It's actually a famous identity and it has a fancy name as well...it is called the Sophie Germain's Identity.
As the name suggests, Sophie Germain's identity was first discovered by Sophie Germain.
Marie-Sophie Germain born on 1 April 1776, she was a French mathematician, physicist, and philosopher.
Sophie Germain made great contributions to the number theory, acoustics and elasticity. Sophie's education was disorgarnized and haphazard. She never received any professional training in any subject. She wanted to study in the Ecole Polytechnique, but was not allowed to enter because she was a woman. Sophie, determined that she would have the chance to join in this male-dominated sphere, came up with a cunning way around the rule. She assumed the identity of a former student, Monsieur le Blanc, and began writing to the lecturer Joseph-Louis Lagrange with solutions to the mathematical problems that he posed to his students. Lagrange was enormously impressed at the talent of this mysterious pupil, and eventually insisted on being able to meet ‘him.’ It was then that Germain had to reveal her true identity.
She used the same trick later on, when she began writing letters to the mathematician Carl Gauss. She impressed him with her mathematical knowledge, and even proposed a mathematical proof which influenced the solving of Fermats last theorem. When Gauss finally discovered that his penpal was a woman, he was so impressed that he wrote, “without doubt she must have the noblest courage, quite extraordinary talents and superior genius.”
But she thrived in complete intellectual isolation. She happened to be the first woman not related to a member to attend the sessions of the French Academy of Sciences. She was alo the first woman to be invited by the Institute of France to attend its sessions. She was successful in making partial progress on a proof of Fermat's Last Theorem.
Germain mathematically solved the problem of Chladini figures, patterns produced by vibrations.
Source taken from Vigyan Prasar Science Portal:
http://www.vigyanprasar.gov.in
http://www.brightknowledge.org
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