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For example, we can prove by induction that all positive integers of the form 2n − 1 are odd. Let P ( n ) represent " 2 n − 1 is odd": (i) For n = 1 , 2 n − 1 = 2(1) − 1 = 1 , and 1 is odd, since it leaves a remainder of 1 when divided by 2 .
One of the widely used types of impossibility proof is proof by contradiction.In this type of proof, it is shown that if a proposition, such as a solution to a particular class of equations, is assumed to hold, then via deduction two mutually contradictory things can be shown to hold, such as a number being both even and odd or both negative and positive.
One of many examples from algebraic geometry in the first half of the 20th century: Severi (1946) claimed that a degree-n surface in 3-dimensional projective space has at most (n+2 3 )−4 nodes, B. Segre pointed out that this was wrong; for example, for degree 6 the maximum number of nodes is 65, achieved by the Barth sextic , which is more ...
Group (mathematics) Halting problem. insolubility of the halting problem; Harmonic series (mathematics) divergence of the (standard) harmonic series; Highly composite number; Area of hyperbolic sector, basis of hyperbolic angle; Infinite series. convergence of the geometric series with first term 1 and ratio 1/2; Integer partition; Irrational ...
It is sometimes desirable to prove a statement involving two natural numbers, n and m, by iterating the induction process. That is, one proves a base case and an induction step for n, and in each of those proves a base case and an induction step for m. See, for example, the proof of commutativity accompanying addition of natural numbers. More ...
The Riemann Hypothesis. Today’s mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It’s one of the seven Millennium Prize ...
An algebraic solution of a polynomial equation is an expression involving the four basic arithmetic operations (addition, subtraction, multiplication, and division), and root extractions. Such an expression may be viewed as the description of a computation that starts from the coefficients of the equation to be solved and proceeds by computing ...
In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".