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The Avogadro constant, commonly denoted N A [1] or L, [2] is an SI defining constant with an exact value of 6.022 140 76 × 10 23 mol −1 (reciprocal moles). [3] [4] It is defined as the number of constituent particles (usually molecules, atoms, ions, or ion pairs) per mole and used as a normalization factor in the amount of substance in a sample.
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, it is possible to expand the polynomial (x + y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending ...
In additive number theory, Fermat 's theorem on sums of two squares states that an odd prime p can be expressed as: with x and y integers, if and only if. The prime numbers for which this is true are called Pythagorean primes. For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of ...
For example, direct proof can be used to prove that the sum of two even integers is always even: Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b, respectively, for some integers a and b. Then the sum is x + y = 2a + 2b = 2(a+b). Therefore x+y has 2 as a factor and, by definition, is even. Hence ...
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. [1][2] The integers 2 3 and 3 2 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose ...
If x = y, then 2x 3 = −z 3, which implies that x is even, a contradiction. Since x and y are both odd, their sum and difference are both even numbers 2u = x + y 2v = x − y. where the non-zero integers u and v are coprime and have different parity (one is even, the other odd). Since x = u + v and y = u − v, it follows that −z 3 = (u + v ...
Morphism of algebraic varieties. In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called ...
Quadratic formula. The roots of the quadratic function y = 1 2 x2 − 3x + 5 2 are the places where the graph intersects the x -axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.