## Functiones et Approximatio Commentarii Mathematici

### Primes in Certain Arithmetic Progressions

#### Abstract

We discuss to what extent Euclid's elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet's theorem). Murty had shown earlier that such proofs can exist if and only if the residue class (mod $k$) has order 1 or 2. After reviewing this work, we consider generalizations of this question to algebraic number fields.

#### Article information

Source
Funct. Approx. Comment. Math. Volume 35, Number 1 (2006), 249-259.

Dates
First available in Project Euclid: 16 December 2008

http://projecteuclid.org/euclid.facm/1229442627

Mathematical Reviews number (MathSciNet)
MR2271617

Zentralblatt MATH identifier
05784932

Digital Object Identifier
doi:10.7169/facm/1229442627

#### Citation

Murty, Ram; Thain, Nithum. Primes in Certain Arithmetic Progressions. Functiones et Approximatio Commentarii Mathematici 35 (2006), no. 1, 249--259. doi:10.7169/facm/1229442627. http://projecteuclid.org/euclid.facm/1229442627.

#### References

• M. Bauer, Zur Theorie der algebraischen Zahlkoerper, Math. Annalen, 77 (1916), 353--356.
• P. Bateman and M.E. Low, Prime numbers in arithmetic progression with difference 24, Amer. Math. Monthly, 72 (1965), 139--143.
• M.R. Murty, On the existence of Euclidean proofs'' of Dirichlet's theorem on primes in arithmetic progressions, B. Sc. Thesis, 1976, (unpublished) Carleton University.