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.
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