Carl Friedrich Gauss
Gauss attended the University of Gottingen from 1795 to 1798. His most prolific year was 1796. On 30 March, he discovered that a 17-sided regular polygon can be constructed using a straightedge and compass. Previously, constructions were known for only regular polygons with 3, 5, or 15 sides, as well as any of these multiplied by a power of 2. Gauss showed that a construction exists for 17-sided polygons as well as 257-gons and 65537-gons. He also ruled out any such construction of many other polygons, such as heptagons. Then, soon after, he proved a very important result known as the quadratic reciprocity law. Given two odd primes p and q, it concerns whether x exists such that x^{2}=q mod p for any integer x. The law states that x^{2}=q mod p for some integer x if and only if there exists an integer y such that y^{2}=p mod q, unless both p=3 mod 4 and q=3 mod 4, in which case x^{2}=q mod p for some integer x if and only if there is no integer y such that y^{2}=p mod q. Also, on 10 July, he discovered that every natural number is the sum of at most three triangular numbers. He then wrote "Eureka! num = Δ + Δ + Δ".
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