xviii Frequently used Theorems and Conjectures

Hypothesis H (or Schinzel Hypothesis)

Let ≥ 1 and f1(x), . . . , f (x) be irreducible polynomials with integer

coeﬃcients and positive leading coeﬃcient. Assume that there are no

integers 1 dividing the product f1(n) . . . f (n) for all positive integers

n. Then there exist infinitely many positive integers m such that all

numbers f1(m), . . . , f (m) are primes.

This conjecture was first stated in 1958 by A. Schinzel and W. Sierpinski [181].

The abc Conjecture

Let ε 0. There exists a positive constant M = M(ε) such that, given

any co-prime integers a, b, c verifying the conditions 0 a b c and

a + b = c, we have

c M ·

⎛

⎝

p|abc

p⎠

⎞1+ε

.

An equivalent statement is the following:

Let ε 0. There exists a positive constant M = M(ε) such that, given

any co-prime integers a, b, c satisfying a + b = c, we have

max{|a|, |b|,|c|} M ·

⎛

⎝

p|abc

p⎠

⎞1+ε

.

The abc Conjecture was first stated in 1985 by D.W. Masser and J. Oesterl´e.