Disquistiones Arithmeticae was published in Leipzig in 1801 and centred on Gauss’s work in number theory. Disqu. Arithm. is divided into seven chapters, called sections. The first three are introductory, sections IV – VI form the central part of the work and section VII is a short monograph devoted to a separate but related subject. The work is dedicated to the Duke of Brunswick.

The first section, only five pages long deals with elementary results and concepts, such as the derivation of the divisibility rules for 3, 9 and 11. As the most basic concept in the work, congruences for rational integers modulo a natural number are defined and their elementary properties are proved, among them the division algorithm.

In section II (24 pages) Gauss proves the uniqueness of the factorisation of integers into primes and defines the concepts of greatest common divisor and least common multiple. After defining the expression a b mod c, Gauss turns to the equation ax + k c. He derives an algorithm for its solution and mentions the possibility of using continued fractions instead if the Euclidean algorithm.

Section III (35 pages) contains an investgation of the residues of a power of a given number modulo (odd) primes. The basis of the investigation is Fermat's ‘little’ theorem

a ^{p - 1} 1 (mod p), p a prime which does not divide a.

Gauss gives two proofs, one by ‘exhaustion’ which goes back to Euler or possibly Leibniz and the other uses the ‘binomial theorem’

In section IV the main topic is the law of quadratic reciprocity. The law derives its name from a formalism invented by Legendre as is defined as follows: let p and q be positive, odd primes. Then

The law of quadratic reciprocity is the identity:

This result had been formulated by Euler and discussed at length by Legendre but had not been proved correctly.

Section V is the main section of Disquistiones Arithmeticae. It deals with the theory of binary quadratic forms ie algebraic expressions of the type:

for given integers a, b and c. A substantial amount of the fifth section is not original but merely summarises the work of Legendre. Gauss clearly indicates where his original work begins and gives credit to others were required. In section VI Gauss presents several important applications of the concepts included in Section V. The principal topics are partial fractions, periodic decimals and the resolution of congruences. Another interesting topic is the derivation of criteria for distinguishing between prime and composite numbers. The remaining section of Disquistiones Arithmeticae deals with ruler and compass constructions and will be discussed shortly.

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