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"Describe" can mean various things: give an algorithm to generate all representations, a closed formula for the number of representations, or even just determine whether any representations exist. The examples above discuss the representation problem for the numbers 3 and 65 by the form x 2 + y 2 {\displaystyle x^{2}+y^{2}} and for the number 1 ...
The roots of the quadratic function y = 1 / 2 x 2 − 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.
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]
Equivalent statement 4 – connection to elliptic curves: If a, b, c is a non-trivial solution to a p + b p = c p, p odd prime, then y 2 = x(x − a p)(x + b p) will be an elliptic curve without a modular form. [17] Examining this elliptic curve with Ribet's theorem shows that it does not have a modular form.
Integrals involving s = √ x 2 − a 2. Assume x 2 > a 2 (for x 2 < a 2, see next section):
Every positive rational number d can be written in the form d = s 2 (t 3 – 91t – 182) for s and t in . For every rational number t, the elliptic curve given by y 2 = x(x 2 – 49(1 + t 4) 2) has rank at least 1. There are many more examples for elliptic curves over number fields.
The sequence () is decreasing and has positive terms. In fact, for all : >, because it is an integral of a non-negative continuous function which is not identically zero; + = + = () () >, again because the last integral is of a non-negative continuous function.
Using this formula to evaluate () at one of the nodes will result in the indeterminate /; computer implementations must replace such results by () =. Each Lagrange basis polynomial can also be written in barycentric form: