A Sieve Method for Generating 1+1 Prime Pairs and Goldbach Function
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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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2026
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| _version_ | 1866902334312808448 |
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| author | Chern, Geeng-Chuan |
| author_facet | Chern, Geeng-Chuan |
| contents | <p><span> </span>A 2-step sieve method for generating <strong>1+1</strong> prime pairs, satisfying Goldbach's conjecture that every even integer greater than 2 can be written as the sum of two primes, has been disclosed. By direct counting based on indication of the Chinese remainder theorem, the so-called Goldbach function, <span>G</span><span>(2</span><span>n</span><span>), </span>which is the number of distinct ways <span>2</span><span>n</span> can be expressed as the sum of two prime numbers,<span> for a positive integer</span><span> n</span>is generated accordingly. The result is<span> </span></p> <p><span><span> </span>G</span><span>(2</span><span>n</span><span>) ~ (</span><span>n</span><span>-1)[(3-2)/(3-1)][(5-2)/(5-1)]… [(</span><span>p<sub>x</sub></span><span>-2)/(</span><span>p<sub>x</sub></span><span>-1)](1-1/2)(1-1/3)(1-1/5)… [1-1/(</span><span>p<sub>y</sub></span><span>)] x </span><span>F<sub>cg</sub></span><span>(</span><span>n</span><span>), where </span><span>p<sub>x</sub></span><span> ~ </span><span>n</span><sup><span>1/2</span></sup><span>, </span><span>p<sub>y</sub></span><span> ~ (2</span><span>n</span><span>)<sup>1/2</sup>, and</span><span> F<sub>cg</sub></span><span>(2</span><span>n</span><span>) is a function defined as follows:</span></p> <p><span><span> </span>F<sub>cg</sub></span><span>(2</span><span>n</span><span>)=1, if there is no prime number</span><span> p</span><span>, 2<</span><span>p</span><u><span><</span></u><span>p<sub>x</sub></span><span>, such that </span><span>n</span><u><span>=</span></u><span> 0 (mod </span><span>p</span><span>);</span></p> <p><span><span> </span>F<sub>cg</sub></span><span>(2</span><span>n</span><span>) = [(</span><span>p<sub>a</sub></span><span>-1)/(</span><span>p<sub>a</sub></span><span>-2)] [(</span><span>p<sub>b</sub></span><span>-1)/(</span><span>p<sub>b</sub></span><span>-2)][(</span><span>p<sub>c</sub></span><span>-1)/(</span><span>p<sub>c</sub></span><span>-2)]….., if there are distinct prime numbers </span><span>p<sub>a</sub></span><span>, </span><span>p<sub>b</sub></span><span>,</span><span> p<sub>c</sub></span><span>,…, 2<</span><span>p<sub>a</sub></span><span><</span><span>p<sub>b</sub></span><span><</span><span>p<sub>c</sub></span><span><…<u><</u></span><span>p<sub>x</sub></span><span>, such that </span><span>n</span><u><span>=</span></u><span> 0 (mod </span><span>p<sub>a</sub></span><span>), </span><span>n</span><u><span>=</span></u><span> 0 (mod </span><span>p<sub>b</sub></span><span>),</span><span> n</span><u><span>=</span></u><span> 0 (mod </span><span>p<sub>c</sub></span><span>)….</span></p> <p><span> </span>When 2<span>n</span> is very large,</p> <p><span><span> </span>G</span><span>(2</span><span>n</span><span>) ~ 2</span><span>CF<sub>cg</sub></span><span>(2</span><span>n</span><span>) x </span><span>n</span><span>/[ln(2</span><span>n</span><span>)]<sup>2</sup></span></p> <p><span>where </span><span>C</span><span> ~ 0.660162 is Hardy–Littlewood's twin prime constant. </span></p> <p><span>G</span><span>(2</span><span>n</span><span>) is a local peak when </span><span>n</span><span> contains more different prime factors which are less than </span><span>n</span><sup><span>1/2</span></sup><span>. For example, </span><span>F<sub>cg</sub></span><span>(2</span><span>n</span><span>) ></span><span> F<sub>cg</sub></span><span>(2</span><span>n</span><u><span>+</span></u><span>2) <u>></u> 1, and, thus, </span><span>G</span><span>(2</span><span>n</span><span>) ></span><span> G</span><span>(2</span><span>n</span><u><span>+</span></u><span>2) when </span><span>n</span><span>=</span><span>p<sub>a</sub>p<sub>b</sub>p<sub>c</sub></span><span>…. The peak distribution can explain the so-called Goldbach's comet.<sup>[1]</sup></span></p> <p><span> </span><span>Since </span><span>n</span><span>>>[ln(2</span><span>n</span><span>)]<sup>2 </sup>&</span><span> F<sub>cg</sub></span><span>(2</span><span>n</span><span>)<u>></u>1 =></span><span>G</span><span>(2</span><span>n</span><span>)>>1 for a large 2</span><span>n</span><span> even number, there exists at least a pair of primes such that 2</span><span>n</span><span> is the sum of these two primes. This is also true for every small 2</span><span>n</span><span>>2 by inspection or direct calculation without mathematical approximations. Therefore, e</span><span>very even integer greater than 2 can be written as the sum of two primes.</span></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18291262 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Sieve Method for Generating 1+1 Prime Pairs and Goldbach Function Chern, Geeng-Chuan Goldbach conjecture Goldbach function Proof of 1+1 primes <p><span> </span>A 2-step sieve method for generating <strong>1+1</strong> prime pairs, satisfying Goldbach's conjecture that every even integer greater than 2 can be written as the sum of two primes, has been disclosed. By direct counting based on indication of the Chinese remainder theorem, the so-called Goldbach function, <span>G</span><span>(2</span><span>n</span><span>), </span>which is the number of distinct ways <span>2</span><span>n</span> can be expressed as the sum of two prime numbers,<span> for a positive integer</span><span> n</span>is generated accordingly. The result is<span> </span></p> <p><span><span> </span>G</span><span>(2</span><span>n</span><span>) ~ (</span><span>n</span><span>-1)[(3-2)/(3-1)][(5-2)/(5-1)]… [(</span><span>p<sub>x</sub></span><span>-2)/(</span><span>p<sub>x</sub></span><span>-1)](1-1/2)(1-1/3)(1-1/5)… [1-1/(</span><span>p<sub>y</sub></span><span>)] x </span><span>F<sub>cg</sub></span><span>(</span><span>n</span><span>), where </span><span>p<sub>x</sub></span><span> ~ </span><span>n</span><sup><span>1/2</span></sup><span>, </span><span>p<sub>y</sub></span><span> ~ (2</span><span>n</span><span>)<sup>1/2</sup>, and</span><span> F<sub>cg</sub></span><span>(2</span><span>n</span><span>) is a function defined as follows:</span></p> <p><span><span> </span>F<sub>cg</sub></span><span>(2</span><span>n</span><span>)=1, if there is no prime number</span><span> p</span><span>, 2<</span><span>p</span><u><span><</span></u><span>p<sub>x</sub></span><span>, such that </span><span>n</span><u><span>=</span></u><span> 0 (mod </span><span>p</span><span>);</span></p> <p><span><span> </span>F<sub>cg</sub></span><span>(2</span><span>n</span><span>) = [(</span><span>p<sub>a</sub></span><span>-1)/(</span><span>p<sub>a</sub></span><span>-2)] [(</span><span>p<sub>b</sub></span><span>-1)/(</span><span>p<sub>b</sub></span><span>-2)][(</span><span>p<sub>c</sub></span><span>-1)/(</span><span>p<sub>c</sub></span><span>-2)]….., if there are distinct prime numbers </span><span>p<sub>a</sub></span><span>, </span><span>p<sub>b</sub></span><span>,</span><span> p<sub>c</sub></span><span>,…, 2<</span><span>p<sub>a</sub></span><span><</span><span>p<sub>b</sub></span><span><</span><span>p<sub>c</sub></span><span><…<u><</u></span><span>p<sub>x</sub></span><span>, such that </span><span>n</span><u><span>=</span></u><span> 0 (mod </span><span>p<sub>a</sub></span><span>), </span><span>n</span><u><span>=</span></u><span> 0 (mod </span><span>p<sub>b</sub></span><span>),</span><span> n</span><u><span>=</span></u><span> 0 (mod </span><span>p<sub>c</sub></span><span>)….</span></p> <p><span> </span>When 2<span>n</span> is very large,</p> <p><span><span> </span>G</span><span>(2</span><span>n</span><span>) ~ 2</span><span>CF<sub>cg</sub></span><span>(2</span><span>n</span><span>) x </span><span>n</span><span>/[ln(2</span><span>n</span><span>)]<sup>2</sup></span></p> <p><span>where </span><span>C</span><span> ~ 0.660162 is Hardy–Littlewood's twin prime constant. </span></p> <p><span>G</span><span>(2</span><span>n</span><span>) is a local peak when </span><span>n</span><span> contains more different prime factors which are less than </span><span>n</span><sup><span>1/2</span></sup><span>. For example, </span><span>F<sub>cg</sub></span><span>(2</span><span>n</span><span>) ></span><span> F<sub>cg</sub></span><span>(2</span><span>n</span><u><span>+</span></u><span>2) <u>></u> 1, and, thus, </span><span>G</span><span>(2</span><span>n</span><span>) ></span><span> G</span><span>(2</span><span>n</span><u><span>+</span></u><span>2) when </span><span>n</span><span>=</span><span>p<sub>a</sub>p<sub>b</sub>p<sub>c</sub></span><span>…. The peak distribution can explain the so-called Goldbach's comet.<sup>[1]</sup></span></p> <p><span> </span><span>Since </span><span>n</span><span>>>[ln(2</span><span>n</span><span>)]<sup>2 </sup>&</span><span> F<sub>cg</sub></span><span>(2</span><span>n</span><span>)<u>></u>1 =></span><span>G</span><span>(2</span><span>n</span><span>)>>1 for a large 2</span><span>n</span><span> even number, there exists at least a pair of primes such that 2</span><span>n</span><span> is the sum of these two primes. This is also true for every small 2</span><span>n</span><span>>2 by inspection or direct calculation without mathematical approximations. Therefore, e</span><span>very even integer greater than 2 can be written as the sum of two primes.</span></p> |
| title | A Sieve Method for Generating 1+1 Prime Pairs and Goldbach Function |
| topic | Goldbach conjecture Goldbach function Proof of 1+1 primes |
| url | https://doi.org/10.5281/zenodo.18291262 |