In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The m...In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The method that we use is to analyze binomial coefficients. It is developed by the author from the method of analyzing binomial central coefficients, that was used by Paul Erdős in his proof of Bertrand’s postulate - Chebyshev’s theorem.展开更多
The paper contains a geometric interpretation of Andrica’s conjecture about the gap between the square roots of the consecutive primes and brings empirical evidence that the random fluctuations of the gap between the...The paper contains a geometric interpretation of Andrica’s conjecture about the gap between the square roots of the consecutive primes and brings empirical evidence that the random fluctuations of the gap between the quare roots of the consecutive primes seem to stabilize around the mean gap.展开更多
文摘In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The method that we use is to analyze binomial coefficients. It is developed by the author from the method of analyzing binomial central coefficients, that was used by Paul Erdős in his proof of Bertrand’s postulate - Chebyshev’s theorem.
文摘The paper contains a geometric interpretation of Andrica’s conjecture about the gap between the square roots of the consecutive primes and brings empirical evidence that the random fluctuations of the gap between the quare roots of the consecutive primes seem to stabilize around the mean gap.
