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• W1569168556 abstract "We develop an asymptotic prime counting function based on the sieve process of Erathostenes. This function can be generalized to include a finite set of conditions imposed on sums (generalized Goldbach conditions) or differences (generalized Polignac conditions) of M primes. With one additive condition for two primes p1, p2 (M=2) with p1+p2=E, the generalized counting function approximates the number of Goldbach partitions and its average lower bound. Its relative errors decrease with increasing even number E, and so enable it to explain the structure of the Goldbach-pair counting function. With one subtractive condition for two primes (M=2), the number of prime pairs with a given distance (e.g. twin primes with distance 2) in the interval 2...Z (Z=integer) can be estimated. For more than two primes (M>2), additive and subtractive conditions can be mixed. As examples, counting functions are considered to investigate strings of M=5 or more equidistant primes, twin-Goldbach pairs (M=4), and well known triplets (M=3) or quadruplets (M=4). The generalized counting function is based on an equality assumption we cannot prove (a would be equivalent to a simultaneous for the Goldbach conjecture and other conjectures involving more than 2 primes). However, reasons given for its validity include an observed scaling relation for the quasi-random errors of the generalized counting function, leading to a heuristic proof for the Goldbach conjecture. The problem Instead of trying to solve one of the hardest problems in number theory, we make the task even more difficult! Beyond asking for at least one prime pair adding up to a given even number E, we are e.g. interested in the number of prime pairs p1+p2=E that are at the same time members of twin primes in such a way that p1-2 and p2+2 are also primes. For E=36 the three twin-Goldbach pairs would be 7+29, 19+17 and 31+5. For which E≥12 are there solutions and how many? Our generalized counting function answers these and other questions with an accuracy quickly increasing with increasing E. Does our method show a way heading towards a of the Goldbach conjecture? Introduction Goldbach's conjecture from 1742 states that every even integer E>2 can be expressed as sum of two primes (in the following called Goldbach pairs or Goldbach partitions). Though computer-based numerical tests have confirmed the theorem up to 4·10 (and ongoing computations have increased this limit to over 8·10), it resisted any formal mathematical up to now. Polignac's conjecture from 1849 states that there exist infinitely many pairs of consecutive primes with any given even difference D≥2. A necessary condition for the conjectures to be valid is, that" @default.
• W1569168556 created "2016-06-24" @default.
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• W1569168556 date "2007-01-01" @default.
• W1569168556 modified "2023-09-23" @default.
• W1569168556 title "Generalization of the Goldbach-Polignac conjectures and estimation of counting functions" @default.
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• W1569168556 doi "https://doi.org/10.3929/ethz-a-005421216" @default.
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