In this research proof of Legendre conjecture is presented. The proof is based on a property possessed exclusively by all prime numbers. That is, the positive square-root of any prime number is an irrational number that always lies between two consecutive positive integers. This property excludes the number one from the set of prime numbers. Not all composite numbers possess this sure property possessed by all prime numbers. It is this special property of prime numbers special property of prime numbers that makes Legendre conjecture a sure law for all prime numbers.
In the process of seeking to prove Legendre’s conjecture the prime gap problem is resolved and Riemann hypothesis reviewed in the light of these findings.
A few methods are known for the measurements of bended flat areas. But the problems arise when the areas are bended with many bends and deflections. In this situation we have to depend on Simpson’s rule. If we follow this rule however, we can measure areas with certain limitations. Therefore accurate results are not always achieved. Recently a theorem has been reported which is based on conversion of gravitational weight of a map occupied by the area of the flat surface. This procedure appears to be simple, prompt and accurate. However a comparative study of area measurements is therefore needful to determine the validity of this theorem.
In the paper the question of the existence of a differential inclusion under the initial condition is considered. It is assumed that a multivalued mapping a continuous and the sets are almost convex.
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