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Generalizing a theorem of Schur

Lenny Jones (2014)

Czechoslovak Mathematical Journal

In a letter written to Landau in 1935, Schur stated that for any integer t > 2 , there are primes p 1 < p 2 < < p t such that p 1 + p 2 > p t . In this note, we use the Prime Number Theorem and extend Schur’s result to show that for any integers t k 1 and real ϵ > 0 , there exist primes p 1 < p 2 < < p t such that p 1 + p 2 + + p k > ( k - ϵ ) p t .

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