an:00727789
Zbl 0815.11044
Ageev, A. A.
Sierpinski's theorem is deducible from Euler and Dirichlet
EN
Am. Math. Mon. 101, No. 7, 659-660 (1994).
00022236
1994
j
11N32 11N13 11A41
primes represented by polynomials; Dirichlet's theorem; arithmetic progression; Euler's result; sum of two squares
In 1964 \textit{W. Sierpi??ski} [Bull. Soc. R. Sci. Li??ge 33, 259-260 (1964; Zbl 0127.268)] proved that for any \(M\) there exists a positive integer \(t\) such that the sequence \(n^ 2 + t\), \(n = 1,2, \dots\) contains at least \(M\) primes. In the present note the author shows that an even slightly stronger result can be easily derived from Dirichlet's theorem that every arithmetic progression with common difference relatively prime to the initial term contains infinitely many primes in conjunction with Euler's result that every prime of the form \(4k + 1\) is representable as a sum of two squares.
J.Hinz (Marburg)
Zbl 0127.268