Prove-that-sum-of-all-prime-numbers-p-such-that-n-p-2-k-n-is-2-k-n-n-p-2-k-n-p-2-k-n-p-prime-

Tinku Tara June 3, 2023 Number Theory 0 Comments

Question Number 3630 by prakash jain last updated on 16/Dec/15

Prove that sum of all prime numbers p such that n≤p≤2^k n is ≥2^k n. Σ_(n≤p≤2^k n) p ≥ 2^k n (p−prime)

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{prime}\:\mathrm{numbers}\:{p} \\ $$$$\mathrm{such}\:\mathrm{that}\:{n}\leqslant{p}\leqslant\mathrm{2}^{{k}} {n}\:\mathrm{is}\:\geqslant\mathrm{2}^{{k}} {n}. \\ $$$$\underset{{n}\leqslant{p}\leqslant\mathrm{2}^{{k}} {n}} {\sum}{p}\:\:\geqslant\:\mathrm{2}^{{k}} {n}\:\:\:\:\:\left({p}−{prime}\right) \\ $$

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Prove-that-sum-of-all-prime-numbers-p-such-that-n-p-2-k-n-is-2-k-n-n-p-2-k-n-p-2-k-n-p-prime-

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