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Prime Number Theorem; Schur
In a letter written to Landau in 1935, Schur stated that for any integer $t>2$, there are primes $p_{1}<p_{2}<\cdots <p_{t}$ 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 $t\ge k \ge 1$ and real $\epsilon >0$, there exist primes $p_{1}<p_{2}<\cdots <p_{t}$ such that \[ p_{1}+p_{2}+\cdots +p_{k}>(k-\epsilon )p_{t}. \]
[1] Lehmer, E.: On the magnitude of the coefficients of the cyclotomic polynomial. Bull. Am. Math. Soc. 42 389-392 (1936). DOI 10.1090/S0002-9904-1936-06309-3 | MR 1563307 | Zbl 0014.39203
[2] Suzuki, J.: On coefficients of cyclotomic polynomials. Proc. Japan Acad., Ser. A 63 279-280 (1987). DOI 10.3792/pjaa.63.279 | MR 0931264 | Zbl 0641.10008
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