Abstract
Bu makalede, asal sayıların belirlenmesinde kullanılan bilinen deterministik ve olasılıksal yöntemlerin aksine, sadece cebirsel analize dayalı yeni bir deterministik yöntem türü, tasarlanmış bir denklem oluşturmadan ve ön varsayımlar ve önkoşullar yazmadan kanıtlanacaktır. Bunun için mod 30'da sayıların katmanlarında 30k + 1, 30k + 7, 30k + 11, 30k + 13, 30k + 17, 30k + 19, 30k + 23, 30k + yerlerinin olduğu kanıtlanacaktır. İlk üçü dışındaki asal sayıların bulunduğu yer, kendi içinde kapalı bir sistem oluşturur. Bu sekiz konum sekiz katman olarak anılacak ve ayrıca asal sayıların kapalı sistemde sekiz katmana eşit olarak dağıtıldığı (eşit dağıtım ilkesi) açıklanacaktır. Bu yeni yöntemle, işlem yükünü önemli ölçüde azaltma olasılıkları olduğu da gösterilecektir.
Keywords
Sekiz katmandaki asal sayılar Kapalı sistem Eşit dağılım Önkoşulsuz yeni deterministik cebirsel yöntem
References
- 1. Agrawal M, Kayal N, Saxena N. Primes is in P., Ann. Math., 2004;160:781-793.
- 2. Miller GL. Riemann’s hypothesis and tests for primality, J. Comput. Sys. Sci., 1976;13:300-317.
- 3. Rabin MO. Probabilistic algoritm for testing primality, J. Number Theory, 1980;12:128-138.
- 4. Solovay R., Strassen V. A fast Monte-Carlo test for primality, SIAM Journal on Computing, 1977;6:84-86.
- 5. Adleman LM, Pomerance C, Rumely RS. On distinguishing prime numbers from composite numbers, Ann. of Math., 1983;117:173-206.
- 6. Goldwasser S, Kilian J. Almost all primes can be quickly certified, Proc. 18th STOC., 1986; 316-329.
- 7. Adleman LM, Huang MD. Primality testing and two dimensional Abelian varieties over finite fields, Lecture Notes in Mathematics, 1992;1512.
- 8. Burthe RJJr. Further investigations with the strong probable prime test, Mathematics of Computation, 1996;65(213):373-381.
- 9. Arnault F. Rabin-Miller primality test: Composite numbers which pass it, Mathematics of Computation, 1995;64(209):355-361.
- 10. Gratham J. A probable prime test with high confidence, J. Number Theory, 1998; 72: 32-47.
- 11. Blum M. A machine-independent theory of the complexity of recursive functions, J. Assoc. Comput. Mach., 1967;14: 332-336.
- 12. Rogers HJr. Theory of recursive functions and effective computability. New York: McGraw-Hill; 1967. MR37#61.
- 13. Borodin A. Computational complexity and the existence of complexity gaps, J. Assoc. Comput. Mach., 1972;19:158-174.
- 14. https://mathworld.wolfram.com/RiemannZetaFunction.html. September 10, 2020.
Abstract
In this article, unlike the known deterministic and probabilistic methods used in determining prime numbers, a new type of deterministic method based on only algebraic analysis will be proven without creating a designed equation and without writing preliminary assumptions and prerequisites. For this, in the layers of the numbers in mod 30, it will be proven that the places 30k+1, 30k+7, 30k+11, 30k+13, 30k+17, 30k+19, 30k+23, 30k+29 where the prime numbers except for the first three locate, form a closed system in themselves. These eight locations will be referred to as eight layers and it will also be explained that the prime numbers are distributed equally across eight layers (equidistribution principle) in the closed system. With this new method, it will also be shown that there are possibilities to reduce the processing load considerably.
Keywords
Prime numbers in eight layers closed system equal distribution unconditional new deterministic algebraic method
References
- 1. Agrawal M, Kayal N, Saxena N. Primes is in P., Ann. Math., 2004;160:781-793.
- 2. Miller GL. Riemann’s hypothesis and tests for primality, J. Comput. Sys. Sci., 1976;13:300-317.
- 3. Rabin MO. Probabilistic algoritm for testing primality, J. Number Theory, 1980;12:128-138.
- 4. Solovay R., Strassen V. A fast Monte-Carlo test for primality, SIAM Journal on Computing, 1977;6:84-86.
- 5. Adleman LM, Pomerance C, Rumely RS. On distinguishing prime numbers from composite numbers, Ann. of Math., 1983;117:173-206.
- 6. Goldwasser S, Kilian J. Almost all primes can be quickly certified, Proc. 18th STOC., 1986; 316-329.
- 7. Adleman LM, Huang MD. Primality testing and two dimensional Abelian varieties over finite fields, Lecture Notes in Mathematics, 1992;1512.
- 8. Burthe RJJr. Further investigations with the strong probable prime test, Mathematics of Computation, 1996;65(213):373-381.
- 9. Arnault F. Rabin-Miller primality test: Composite numbers which pass it, Mathematics of Computation, 1995;64(209):355-361.
- 10. Gratham J. A probable prime test with high confidence, J. Number Theory, 1998; 72: 32-47.
- 11. Blum M. A machine-independent theory of the complexity of recursive functions, J. Assoc. Comput. Mach., 1967;14: 332-336.
- 12. Rogers HJr. Theory of recursive functions and effective computability. New York: McGraw-Hill; 1967. MR37#61.
- 13. Borodin A. Computational complexity and the existence of complexity gaps, J. Assoc. Comput. Mach., 1972;19:158-174.
- 14. https://mathworld.wolfram.com/RiemannZetaFunction.html. September 10, 2020.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Publication Date | June 30, 2021 |
| Submission Date | January 2, 2021 |
| Acceptance Date | June 8, 2021 |
| Published in Issue | Year 2021 Volume: 5 Issue: 1 |