Example of the use of observation in pure mathematics
According to Jacobi, a treatise with this title was read to the Berlin Academy on November 22, 1753. According to St. Petersburg Academy records, it was presented to that Academy on September 30, 1754.
Euler attempts to explain his thought process by working out problems in quadratic forms in great detail. Euler notes that numbers of the form 2aa + bb are closed under multiplication. He develops more properties of such numbers and proves some of them. Then he uses these properties to show that 67579 = 2∙872+2292 is prime, while 40081 is not.
Original Source Citation
Novi Commentarii academiae scientiarum Petropolitanae, Volume 6, pp. 185-230.
Opera Omnia Citation
Series 1, Volume 2, pp.459-492.