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Sierpinski's Theorem is Deducible from Euler and Dirichlet

Summary: Sierpinski's Theorem is Deducible from
Euler and Dirichlet
A. A. Ageev
The famous theorem of Dirichlet says that the sequence
an + b; n = 1; 2; : : :
contains in nitely many primes if a and b are relatively prime integers. It is a similar
long-standing conjecture that the quadratic sequence
n 2 + t; n = 1; 2; : : :
contains in nite number of primes for any positive integer t. The rst result in this
direction is due to Sierpinski [6] who showed that for any M there exists t 0 such that the
n 2 + t 0 ; ; n = 1; 2; : : :
contains at least M primes. Recently several new proofs and extensions of Sierpinski's
theorem have been appeared [4][3][1].
In this note we show that a slightly stronger result can be easily derived from the
following well-known number theory facts:
Theorem 1 (Euler) ([5, Theorem 251]) Every prime of the form 4n+1 is representable
as a sum of two squares. This representation is unique up to the order of the summands.
Theorem 2 (Dirichlet) ([2, p. 34]) The series


Source: Ageev, Alexandr - Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk


Collections: Mathematics