Computational Problems, Methods, and Results in Algebraic by Horst G Zimmer

By Horst G Zimmer

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Bredihin {1962,1963a) who later (Bredihin 1963b,1964) established the infinity of primes of the form 1p(x,y) + c, where 1p(x,y) is a quadratic form whose discriminant is not a square and c is a non-zero fixed integer (see also Motohashi 1970a,1971). More generally, 95 Bouniakowsky, Victor (1804-1889), Professor in Sankt Petersbourg. 42 1. Early Times every quadratic polynomial in two variables with integral coefficients, which satisfies certain necessary conditions represents infinitely many primes, as shown in lwaniec (1972a,b,1973/74).

There is no non-constant rational function R(X) with integral coefficients which assumes prime values at all sufficiently large integers. Proof. Let R(X) = f(X)fg(X), where J,g are relatively prime polynomials with integral coefficients and assume that for n = N, N + 1, ... one has R(n) = qn, where {qn} is a sequence of primes. Assume also that R(X) is non-constant. By increasing N, if necessary, we may assume that the primes qn (n ~ N) are all distinct. There exist polynomials A(X), B(X) E Z[X] and a non-zero integer c such that A(X)J(X) + B(X)g(X) = c holds.

Essentially a method of exclusion, by which all composite numbers are succesively erased from the series of natuml numbers, and the primes alone are left remaining. It requires only one kind of arithmetic opemtion; that is to say, the formation of consecutive multiples of given numbers, or, in other words, addition only. Legendre (1830). 333 of English translation). g. the existence of primes between x and its double. '(n) = { (-i)' ifn = 1 if n is a product of k distinct primes otherwise. Mobius 70 (1832), who denoted it by bn.

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