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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Noam Elkies was first to find an infinite series of solutions to the Euler''s equation with just one variable equal zero, thus disproving the Euler''s sum of powers conjecture for the fourth power. However, until Jacobi and Madden''s publication, it was not known whether there exist infinitely many solutions to the Euler''s equation with all variables non-zero. Only a finite number of such solutions was known.