Brun was born at Lier in Buskerud, Norway. He studied at the University of Oslo and began research at the University of Göttingen in 1910. In 1923, Brun became a professor at the Technical University in Trondheim and in 1946 a professor at the University of Oslo. 
In 1915, he introduced a new method, based on Legendre's version of the sieve of Eratosthenes, now known as the Brun sieve, which addresses additive problems such as Goldbach's conjecture and the twin prime conjecture. He used it to prove that there exist infinitely many integers n such that n and n+2 have at most nine prime factors, and that all large even integers are the sum of two numbers with at most nine prime factors.
He also showed that the sum of the reciprocals of twin primes converges to a finite value, now called Brun's constant: by contrast, the sum of the reciprocals of all primes is divergent. He developed a multi-dimensional continued fraction algorithm in 1919–1920 and applied this to problems in musical theory. He also served as praeses of the Royal Norwegian Society of Sciences and Letters in 1946.
- "Viggo Brun". numbertheory.org. 18 June 2003. Retrieved January 1, 2017.
- "Viggo Brun". Store norske leksikon. Retrieved January 1, 2017.
- J J O'Connor and E F Robertson. "Viggo Brun". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved January 1, 2017.
- Bratberg, Terje (1996). "Vitenskapsselskapet". In Arntzen, Jon Gunnar. Trondheim byleksikon. Oslo: Kunnskapsforlaget. pp. 599–600. ISBN 82-573-0642-8.
- Bent Birkeland. "Viggo Brun". Norsk biografisk leksikon. Retrieved January 1, 2017.
- H. Halberstam and H. E. Richert, Sieve methods, Academic Press (1974) ISBN 0-12-318250-6. Gives an account of Brun's sieve.
- C.J. Scriba, Viggo Brun, Historia Mathematica 7 (1980) 1-6.
- C.J. Scriba, Zur Erinnerung an Viggo Brun, Mitt. Math. Ges. Hamburg 11 (1985) 271-290
- Brun's Constant
- Brun's Pure Sieve
- Viggo Brun personal archive exists at NTN University Library Dorabiblioteket
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