Research on the number of prime numbers between n² and (n+1)²

Let pᵣ₊₁-1>n ≥ pᵣ-1, based on a sequence \1,2,3⋯\ Mᵣ(Mᵣ=p₁p₂⋯ pᵣ)\, we compare the density of coprime numbers and establish a correlation between the proportions of coprime numbers in the ranges from 1 to consecutive square numbers. Then, we derive the relationship between the number of coprimes in the interval of n² ~ (n+1)² and the proportion of coprimes in the interval of 1 ~ n², proving that there is at least one prime number between any n² and (n+1)². By extending our research to the range of 1 ~ Mᵣ², we establish the relationship between the proportions of backwards coprime numbers in the ranges from Mᵣ² to consecutive square numbers; furthermore, we establish a relationship between the proportions of coprimes in small interval and the whole interval. Then, in conclusion, the number of coprimes between n² and (n+1)² is greater than n∏ᵢ₌₁ʳ(1-1/pᵢ), thus proving that there are at least 2 prime numbers between n² and (n+1)².