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[parent] example of failure function (polynomials) (Example)

The object of this addition is two-fold:

1) There is nothing better than a numerical example to illustrate a new concept. Let me take the following example: Let $ \phi(x)$ be the polynomial $ x^{2} + x + 1$ ( $ x \in \mathbb{N}$). $ \phi(1) = 3$. Here $ x_0 = 1$. $ \psi (x_0) = 3k +1$ which yields values of $ x = 4,7, 10,\ldots$ such that when substituted in $ \phi(x)$ we get a set of failures ( composite numbers ). We get one additional information: all these failures have 3 as one of the factors.

2) To explain the concept of “second order failure functions”: If we were to substitute the relevant failure function in the original polynomial we get a polynomial in $ k$, viz. $ 9k^{2} + 9k + 3 = 3( 3k^{2} + 3k +1 )$. The expression in parentheses represents the second factor.



"example of failure function (polynomials)" is owned by akdevaraj. [ full author list (2) ]
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Cross-references: represents, expression, viz, function, factors, information, composite numbers, polynomial, addition, object

This is version 5 of example of failure function (polynomials), born on 2008-05-08, modified 2008-05-08.
Object id is 10572, canonical name is ExampleOfFailureFunctionPolynomials.
Accessed 93 times total.

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AMS MSC: 33-01 (Special functions :: Instructional exposition )
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