Essay: Heath Brown’s Equation

Dr. Goldston’s theory is based on Heath-Brown’s equation from theorem 2 for conjecture C1 and Schlage Puchta prove of conjecture C3. They used the prime factorization for integer n as,. They Goldston and his researchers defined this exponent patter for n as the multi set of (Goldston et al., 2010).

Since the researchers were using the functions x and x+1, it obvious the need for an exponential pattern. Therefore, the functions will have values of,  and. They therefore proved that a series did have infinitely many x integers where x and x+1 have exponent values {2, 1, 1, 1}. This leads to theorem 4 in which there exists many x integers such that

,  and

Looking at Dr. Goldston and the researcher’s approach it is evident that they were trying to combine the basic theorems for conjectures to create linear forms (Goldston et al., 2010). The basic theorem used for this research took the form of;

where

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