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Carl Friedrich Gauss
Carl Friedrich Gauss was born on 30 April 1777. This date was not actually
recorded, as Gauss's mother was illiterate and didn't know the date. However,
she did remember that it was on a Wednesday, eight days before the Feast
of Ascension. Later, Gauss himself calculated his date of birth by a formula
for the date of Easter, as the Feast of Ascension is always 40 days after
Easter. As a child, Gauss showed remarkable talent. When he was 3, he
noticed and corrected an error in his father's payroll calculations. In
school, his teacher, J.G.Buttner once set him the task of adding up all
numbers from 1 to 100, which he presumed would be very tedious and would
occupy him for a very long time. However, within seconds he had the answer.
His method was to rearrange the sum to add the numbers in pairs, giving
(1 + 100) + (2 + 99) ... + (50 + 51). Each pair has a sum of 101 and there
are 50 pairs so the answer is 5050.
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 x2=q mod p for any integer x. The law states that x2=q mod p for some integer x if and only if there exists an integer y such that y2=p mod q, unless both p=3 mod 4 and q=3 mod 4, in which case x2=q mod p for some integer x if and only if there is no integer y such that y2=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".