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Note for the Prime Gaps
Abstract
A prime gap is the difference between two successive prime numbers. The nth prime gap, denoted $g_{n}$ is the difference between the (n + 1)st and the nth prime numbers, i.e. $g_{n}=p_{n+1}-p_{n}$. There isn't a verified solution to Andrica's conjecture yet. The conjecture itself deals with the difference between the square roots of consecutive prime numbers. While mathematicians have proven it true for a vast number of primes, a general solution remains elusive. The Andrica's conjecture is equivalent to say that $g_{n}<2 \cdot {\sqrt {p_{n}}}+1$ holds for all $n$. In this note, using the divergence of the infinite sum of the reciprocals of all prime numbers, we prove that the Andrica's conjecture is true.
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References
Jackson Terence. 95.42 Irrational square roots of natural numbers — a geometrical approach. The Mathematical Gazette. Vol. 95(533):327–330. 2011. Cambridge University Press (CUP). [Cross Ref]