Hales-Jewett Teoremi ile Asalların Sonsuzluğu

Öz

Bu çalışmada, asal sayıların sonsuzluğunu kanıtlamak için Hales–Jewett Teoremi kullanılarak farklı bir yol izlenmektedir. Bu yaklaşım, asal sayıların dağılımını incelerken, onların belirli kombinatoriyel yapılar içinde nasıl yer aldığını ortaya koyar. Böylece asal sayıların yalnızca klasik aritmetik yöntemlerle değil, aynı zamanda daha geniş yapısal çerçeveler içinde de ele alınabileceği gösterilmiş olur.

Anahtar Kelimeler

• Asalların sonsuzluğu
• Hales-Jewett Teoremi
• Ramsey teorisi

The Infinitude of the Primes via the Hales-Jewett Theorem

Öz

We take a different approach to prove the infinitude of the prime numbers by utilizing the Hales–Jewett Theorem. This method allows us to examine the distribution of primes while observing how they are situated within specific combinatorial structures. In this way, we demonstrate that primes can be studied not only through classical arithmetic methods but also within broader structural frameworks.

Anahtar Kelimeler

• Hales-Jewett theorem
• Infinitude of the primes
• Ramsey Theory

Kaynakça

1. Ramsey, F. P. (1930). On a problem of formal logic. Proceedings of the London Mathematical Society 30, 264-286.
2. Schur, I. (1916). Über die kongruenz xm + ym ≡ zm (mod p). Jahresbericht der Deutschen Mathematiker-Vereinigung 25.
3. van derWaerden, B. L. (1927). Beweis einer baudetschen vermutung. Nieuw Archief voorWiskunde 15, 212-216.
4. Hales, A.W., & Jewett, R. I. (1963). Regularity and positional games. Transactions of the American Mathematical Society 106, 222–229.
5. Näslund, M. (2013). The Hales-Jewett Theorem and its application to further generalizations of m, n, k-games.
6. Erd˝os, P., & Turán, P. (1936). On some sequences of integers. Journal of the London Mathematical Society 11, 261–264. https://doi.org/10.1112/jlms/s1-11.4.261
7. Roth, K. F. (1953). On certain sets of integers. Journal of the London Mathematical Society, 28, 104–109. https://doi.org/10.1112/jlms/s1-28.1.104
8. Szemerédi, E. (1975). On sets of integers containing no k elements in arithmetic progression. Acta Arithmetica, 27, 199–245.

9. Alpoge, L. (2015). Van der Waerden and the primes. American Mathematical Monthly 122, 784–785. https://doi.org/10.4169/amer.math.monthly.122.8.784
10. Granville, A. (2017). Squares in arithmetic progressions and infinitely many primes. American Mathematical Monthly, 124, 951–954. doi.org/10.4169/amer.math.monthly.124.10.951
11. Elsholtz, C. (2020). Fermat’s Last Theorem implies Euclid’s infinitude of primes. American Mathematical Monthly, 128, 250–257. https://doi.org/10.1080/00029890.2021.1856544
12. Göral, H., Özcan, H. B., & Sertba¸s, D. C. (2022). The Green-Tao Theorem and the infinitude of primes in domains. American Mathematical Monthly, 130, 114–125.
13. Gasarch, W. (2023). Fermat’s Last Theorem, Schur’s Theorem (in Ramsey theory), and the infinitude of the primes. Discrete Mathematics, 346. https://doi.org/10.1016/j.disc.2023.113877
14. Meštrovi´c, R. (2023). Euclid’s Theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.-2022). https://doi.org/10.48550/arXiv.1202.3670
15. Göral, H. (2020). p−adic metrics and the infinitude of primes. Mathematics Magazine 93, 19–22.
16. Göral, H., & Özcan, H. B. (2020). Several novel proofs of the infinitude of primes. Mathematics Student 89, 91–95.
17. Gasarch, W. (2024). A webpage of papers that prove primes infinite using Ramsey Theory. Retrieved from https://www.cs.umd.edu/ gasarch/TOPICS/ramseyprimes/ramseyprimes.html
18. Hales, A. W., & Jewett, R. I. (2009). Regularity and positional games. In Classic Papers in Combinatorics (pp. 320-327). Boston, MA: Birkhäuser Boston.
19. Gouvêa, F. Q. (2020). p-adic Numbers: An Introduction. Third edition, Springer.