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In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is ...
Fermat's theorem on sums of two squares is strongly related with the theory of Gaussian primes.. A Gaussian integer is a complex number + such that a and b are integers. The norm (+) = + of a Gaussian integer is an integer equal to the square of the absolute value of the Gaussian integer.
As Fermat did for the case n = 4, Euler used the technique of infinite descent. [50] The proof assumes a solution ( x , y , z ) to the equation x 3 + y 3 + z 3 = 0 , where the three non-zero integers x , y , and z are pairwise coprime and not all positive.
This includes Fermat's little theorem (generalised by Euler to non-prime moduli); the fact that = + if and only if ; initial work towards a proof that every integer is the sum of four squares (the first complete proof is by Joseph-Louis Lagrange (1770), soon improved by Euler himself [55]); the lack of non-zero integer solutions to ...
Disquisitiones Arithmeticae (Latin for Arithmetical Investigations) is a textbook on number theory written in Latin by Carl Friedrich Gauss in 1798, when Gauss was 21, and published in 1801, when he was 24. It had a revolutionary impact on number theory by making the field truly rigorous and systematic and paved the path for modern number theory.
Fermat number Factor 1732 Euler + 1732 Euler (fully ... gave a full proof of necessity in 1837. The result is known as the Gauss–Wantzel theorem:
In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own. Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together ...
Gauss's lemma is used in many, [3]: Ch. 1 [3]: 9 but by no means all, of the known proofs of quadratic reciprocity. For example, Gotthold Eisenstein [ 3 ] : 236 used Gauss's lemma to prove that if p is an odd prime then