The year 1796 is most productive for both Johann Carl Friedrich Gauss and number theory.

He discovers a construction of the heptadecagon on March 30.

He further advances modular arithmetic, greatly simplifying manipulations in number theory.

On April 8 he becomes the first to prove the quadratic reciprocity law.

This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic.

The prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers.

Gauss also discovers on July 10 that every positive integer is representable as a sum of at most three triangular numbers; he then then jots down in his diary the note: "ΕΥΡΗΚΑ! num = Δ + Δ' + Δ".

On October 1 he publishes a result on the number of solutions of polynomials with coefficients in finite fields, which one hundred and fifty years later will lead to the Weil conjectures.

Gauss was born on April 30, 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents.

His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs thirty-nine days after Easter).

Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years.

He had been christened and confirmed in a church near the school he attended as a child. 

Gauss was a child prodigy.

A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100.

There are many other anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager.

He completed his magnum opus, Disquisitiones Arithmeticae, in 1798, at the age of twenty-one—though it will not be published until 1801.

This work is fundamental in consolidating number theory as a discipline and will shape the field to the present day.

Gauss's intellectual abilities had attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum (now Braunschweig University of Technology), which he had attended from 1792 to 1795, and to the University of Göttingen from 1795 to 1798.

While at university, Gauss independently rediscovers several important theorems.

His breakthrough occurs in 1796 when he shows that a regular polygon can be constructed by compass and straightedge if the number of its sides is the product of distinct Fermat primes and a power of 2.

This is a major discovery in an important field of mathematics; construction problems have occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately leads Gauss to choose mathematics instead of philology as a career.

Gauss is so pleased with this result that he requests that a regular heptadecagon be inscribed on his tombstone.

The stonemason declines, stating that the difficult construction would essentially look like a circle.

Carl Friedrich Gauss publishes his magnum opus, Disquisitiones Arithmeticae.

A textbook of number theory written in Latin in 1798 when Gauss was twenty-one and first published in 1801 when he is twenty-four, it is notable for having a revolutionary impact on the field of number theory as it will not only turn the field truly rigorous and systematic but paves the path for modern number theory.

In this book, Gauss brings together and reconciles results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds many profound and original results of his own.

Giuseppe Piazzi is the discoverer of Ceres, today known as the largest member of the asteroid belt.

On January 1, 1801, Piazzi discovers a "stellar object" that moves against the background of stars.

At first he thinks it is a fixed star, but once he notices that it moves, he becomes convinced it is a planet, or as he calls it, "a new star".

In spite of his assumption that it is a planet, he takes the conservative route and announces it as a comet.

In a letter to astronomer Barnaba Oriani of [Milan] he makes his suspicions known in writing.

He is not able to observe it long enough as it was soon lost in the glare of the Sun.

Unable to compute its orbit with existing methods, the mathematician Carl Friedrich Gauss will develop a new method of orbit calculation that will allow astronomers to locate it again.

After its orbit is better determined, it will be clear that Piazzi's assumption is correct and this object is not a comet but more like a small planet.

Coincidentally, it is also almost exactly where the Titius-Bode law predicted a planet would be.

Piazzi names it "Ceres Ferdinandea," after the Roman and Sicilian goddess of grain and King Ferdinand IV of Naples and Sicily.

The Ferdinandea part will later be dropped for political reasons.

Ceres turns out to be the first, and largest, of the asteroids existing within the asteroid belt.

Ceres is today called a dwarf planet.

George Green, in An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828), introduces several important concepts, among them a theorem similar to the modern Green's theorem, the idea of potential functions as currently used in physics, and the concept of what are now called Green's functions.

A British mathematician and physicist, whose work will never be well known in the mathematical community during his lifetime, Green is the first person to try to explain a mathematical theory of the theories of electricity and magnetism that forms the basis for other scientists such as James Clerk Maxwell, William Thomson, and others.

His work runs parallel to that of the great German mathematician Carl Friedrich Gauss (potential theory).

The single wire telegraph is investigated and developed independently by Andre-Marie Ampére and by by Carl F. Gausss and Wilhelm F. Weber.

Gauss and Weber obtain permission on May 6 to build an electromagnetic telegraph in Göttingen.