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 sequence 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 X Collections: Mathematics