Proofs of Bertrand's Postulate

Explanations 3

• What is Bertrand's Postulate? Joseph Bertrand stated that, for every n>1n>1n>1, there is a prime between nnn and 2n2n2n. That is, for n>1n>1n>1, there is a ppp with n<p<2nn<p<2n…

  By amaterasu, Po-Shen Loh, Kada Williams, Victor Wang

• Prove that for x≥2x\ge 2x≥2, we have ∏p≤xp<4x−1.\prod_{p\le x}p<4^{x-1}.p≤x∏​p<4x−1.

  By Kada Williams

• Prove that the inequality 22n/3>2n2^{\sqrt{2n}/3}> 2n22n​/3>2n holds for n≥450n\ge 450n≥450. (Hint: 2n2n2n is between two square numbers - k2≤2n<(k+1)2k^2\le 2n<(k+1)^2k2≤2n<(k+…

  By Kada Williams