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In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]
11 Further reading. 12 ... Perhaps the numbers most easy to prove irrational are ... This second definition is used in Errett Bishop's proof that the square root of 2 ...
A more recent proof by Wadim Zudilin is more reminiscent of Apéry's original proof, [6] and also has similarities to a fourth proof by Yuri Nesterenko. [7] These later proofs again derive a contradiction from the assumption that ζ ( 3 ) {\displaystyle \zeta (3)} is rational by constructing sequences that tend to zero but are bounded below by ...
Transcendental numbers therefore represent the typical case; even so, it may be extremely difficult to prove that a given number is transcendental (or even simply irrational). For this reason transcendence theory often works towards a more quantitative approach.
A more general proof shows that the mth root of an integer N is irrational, unless N is the mth power of an integer n. [7] That is, it is impossible to express the mth root of an integer N as the ratio a ⁄ b of two integers a and b, that share no common prime factor, except in cases in which b = 1.
Bertrand's postulate and a proof; Estimation of covariance matrices; Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational
This application also invokes the integer root theorem, a stronger version of the rational root theorem for the case when () is a monic polynomial with integer coefficients; for such a polynomial, all roots are necessarily integers (which is not, as 2 is not a perfect square) or irrational. The rational root theorem (or integer root theorem ...
In addition to theorems of geometry, such as the Pythagorean theorem, the Elements also covers number theory, including a proof that the square root of two is irrational and a proof that there are infinitely many prime numbers. Further advances also took place in medieval Islamic mathematics.